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Properties

Label 14-1881e7-1.1-c1e7-0-2
Degree 1414
Conductor 8.331×10228.331\times 10^{22}
Sign 11
Analytic cond. 1.72449×1081.72449\times 10^{8}
Root an. cond. 3.875543.87554
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·4-s + 2·7-s − 6·8-s + 7·11-s + 3·13-s + 4·14-s − 16-s + 7·17-s − 7·19-s + 14·22-s + 18·23-s − 20·25-s + 6·26-s − 4·28-s + 14·29-s − 8·31-s + 4·32-s + 14·34-s − 2·37-s − 14·38-s + 18·41-s − 4·43-s − 14·44-s + 36·46-s + 10·47-s − 28·49-s + ⋯
L(s)  = 1  + 1.41·2-s − 4-s + 0.755·7-s − 2.12·8-s + 2.11·11-s + 0.832·13-s + 1.06·14-s − 1/4·16-s + 1.69·17-s − 1.60·19-s + 2.98·22-s + 3.75·23-s − 4·25-s + 1.17·26-s − 0.755·28-s + 2.59·29-s − 1.43·31-s + 0.707·32-s + 2.40·34-s − 0.328·37-s − 2.27·38-s + 2.81·41-s − 0.609·43-s − 2.11·44-s + 5.30·46-s + 1.45·47-s − 4·49-s + ⋯

Functional equation

Λ(s)=((314⋅117⋅197)s/2 ΓC(s)7 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((314⋅117⋅197)s/2 ΓC(s+1/2)7 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{14} \cdot 11^{7} \cdot 19^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1414
Conductor: 314⋅117⋅1973^{14} \cdot 11^{7} \cdot 19^{7}
Sign: 11
Analytic conductor: 1.72449×1081.72449\times 10^{8}
Root analytic conductor: 3.875543.87554
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (14, 314⋅117⋅197, ( :[1/2]7), 1)(14,\ 3^{14} \cdot 11^{7} \cdot 19^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 26.1134771726.11347717
L(12)L(\frac12) ≈\approx 26.1134771726.11347717
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
11 (1−T)7 ( 1 - T )^{7}
19 (1+T)7 ( 1 + T )^{7}
good2 1−pT+3pT2−5pT3+21T4−p5T5+55T6−39pT7+55pT8−p7T9+21p3T10−5p5T11+3p6T12−p7T13+p7T14 1 - p T + 3 p T^{2} - 5 p T^{3} + 21 T^{4} - p^{5} T^{5} + 55 T^{6} - 39 p T^{7} + 55 p T^{8} - p^{7} T^{9} + 21 p^{3} T^{10} - 5 p^{5} T^{11} + 3 p^{6} T^{12} - p^{7} T^{13} + p^{7} T^{14}
5 1+4pT2+8T3+42pT4+104T5+1496T6+662T7+1496pT8+104p2T9+42p4T10+8p4T11+4p6T12+p7T14 1 + 4 p T^{2} + 8 T^{3} + 42 p T^{4} + 104 T^{5} + 1496 T^{6} + 662 T^{7} + 1496 p T^{8} + 104 p^{2} T^{9} + 42 p^{4} T^{10} + 8 p^{4} T^{11} + 4 p^{6} T^{12} + p^{7} T^{14}
7 1−2T+32T2−62T3+534T4−902T5+5580T6−7976T7+5580pT8−902p2T9+534p3T10−62p4T11+32p5T12−2p6T13+p7T14 1 - 2 T + 32 T^{2} - 62 T^{3} + 534 T^{4} - 902 T^{5} + 5580 T^{6} - 7976 T^{7} + 5580 p T^{8} - 902 p^{2} T^{9} + 534 p^{3} T^{10} - 62 p^{4} T^{11} + 32 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14}
13 1−3T+4pT2−167T3+1464T4−4317T5+27662T6−68781T7+27662pT8−4317p2T9+1464p3T10−167p4T11+4p6T12−3p6T13+p7T14 1 - 3 T + 4 p T^{2} - 167 T^{3} + 1464 T^{4} - 4317 T^{5} + 27662 T^{6} - 68781 T^{7} + 27662 p T^{8} - 4317 p^{2} T^{9} + 1464 p^{3} T^{10} - 167 p^{4} T^{11} + 4 p^{6} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14}
17 1−7T+75T2−292T3+1692T4−3046T5+16149T6−5591T7+16149pT8−3046p2T9+1692p3T10−292p4T11+75p5T12−7p6T13+p7T14 1 - 7 T + 75 T^{2} - 292 T^{3} + 1692 T^{4} - 3046 T^{5} + 16149 T^{6} - 5591 T^{7} + 16149 p T^{8} - 3046 p^{2} T^{9} + 1692 p^{3} T^{10} - 292 p^{4} T^{11} + 75 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14}
23 1−18T+250T2−2456T3+20475T4−139784T5+837561T6−4269484T7+837561pT8−139784p2T9+20475p3T10−2456p4T11+250p5T12−18p6T13+p7T14 1 - 18 T + 250 T^{2} - 2456 T^{3} + 20475 T^{4} - 139784 T^{5} + 837561 T^{6} - 4269484 T^{7} + 837561 p T^{8} - 139784 p^{2} T^{9} + 20475 p^{3} T^{10} - 2456 p^{4} T^{11} + 250 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14}
29 1−14T+207T2−1678T3+13600T4−76898T5+469825T6−2314678T7+469825pT8−76898p2T9+13600p3T10−1678p4T11+207p5T12−14p6T13+p7T14 1 - 14 T + 207 T^{2} - 1678 T^{3} + 13600 T^{4} - 76898 T^{5} + 469825 T^{6} - 2314678 T^{7} + 469825 p T^{8} - 76898 p^{2} T^{9} + 13600 p^{3} T^{10} - 1678 p^{4} T^{11} + 207 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14}
31 1+8T+127T2+546T3+6732T4+24318T5+295517T6+968768T7+295517pT8+24318p2T9+6732p3T10+546p4T11+127p5T12+8p6T13+p7T14 1 + 8 T + 127 T^{2} + 546 T^{3} + 6732 T^{4} + 24318 T^{5} + 295517 T^{6} + 968768 T^{7} + 295517 p T^{8} + 24318 p^{2} T^{9} + 6732 p^{3} T^{10} + 546 p^{4} T^{11} + 127 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14}
37 1+2T+168T2+88T3+12985T4−6784T5+643605T6−553156T7+643605pT8−6784p2T9+12985p3T10+88p4T11+168p5T12+2p6T13+p7T14 1 + 2 T + 168 T^{2} + 88 T^{3} + 12985 T^{4} - 6784 T^{5} + 643605 T^{6} - 553156 T^{7} + 643605 p T^{8} - 6784 p^{2} T^{9} + 12985 p^{3} T^{10} + 88 p^{4} T^{11} + 168 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14}
41 1−18T+330T2−3598T3+38232T4−305214T5+2408210T6−15423692T7+2408210pT8−305214p2T9+38232p3T10−3598p4T11+330p5T12−18p6T13+p7T14 1 - 18 T + 330 T^{2} - 3598 T^{3} + 38232 T^{4} - 305214 T^{5} + 2408210 T^{6} - 15423692 T^{7} + 2408210 p T^{8} - 305214 p^{2} T^{9} + 38232 p^{3} T^{10} - 3598 p^{4} T^{11} + 330 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14}
43 1+4T+121T2+138T3+6656T4−1626T5+349013T6−19992T7+349013pT8−1626p2T9+6656p3T10+138p4T11+121p5T12+4p6T13+p7T14 1 + 4 T + 121 T^{2} + 138 T^{3} + 6656 T^{4} - 1626 T^{5} + 349013 T^{6} - 19992 T^{7} + 349013 p T^{8} - 1626 p^{2} T^{9} + 6656 p^{3} T^{10} + 138 p^{4} T^{11} + 121 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14}
47 1−10T+169T2−498T3+6964T4+13746T5+308671T6+722962T7+308671pT8+13746p2T9+6964p3T10−498p4T11+169p5T12−10p6T13+p7T14 1 - 10 T + 169 T^{2} - 498 T^{3} + 6964 T^{4} + 13746 T^{5} + 308671 T^{6} + 722962 T^{7} + 308671 p T^{8} + 13746 p^{2} T^{9} + 6964 p^{3} T^{10} - 498 p^{4} T^{11} + 169 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14}
53 1−23T+399T2−4428T3+43616T4−334702T5+2604471T6−17887421T7+2604471pT8−334702p2T9+43616p3T10−4428p4T11+399p5T12−23p6T13+p7T14 1 - 23 T + 399 T^{2} - 4428 T^{3} + 43616 T^{4} - 334702 T^{5} + 2604471 T^{6} - 17887421 T^{7} + 2604471 p T^{8} - 334702 p^{2} T^{9} + 43616 p^{3} T^{10} - 4428 p^{4} T^{11} + 399 p^{5} T^{12} - 23 p^{6} T^{13} + p^{7} T^{14}
59 1−15T+376T2−3565T3+51380T4−348761T5+3987278T6−22528211T7+3987278pT8−348761p2T9+51380p3T10−3565p4T11+376p5T12−15p6T13+p7T14 1 - 15 T + 376 T^{2} - 3565 T^{3} + 51380 T^{4} - 348761 T^{5} + 3987278 T^{6} - 22528211 T^{7} + 3987278 p T^{8} - 348761 p^{2} T^{9} + 51380 p^{3} T^{10} - 3565 p^{4} T^{11} + 376 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14}
61 1−12T+362T2−2844T3+50298T4−275584T5+4043948T6−17899330T7+4043948pT8−275584p2T9+50298p3T10−2844p4T11+362p5T12−12p6T13+p7T14 1 - 12 T + 362 T^{2} - 2844 T^{3} + 50298 T^{4} - 275584 T^{5} + 4043948 T^{6} - 17899330 T^{7} + 4043948 p T^{8} - 275584 p^{2} T^{9} + 50298 p^{3} T^{10} - 2844 p^{4} T^{11} + 362 p^{5} T^{12} - 12 p^{6} T^{13} + p^{7} T^{14}
67 1+6T+188T2+2642T3+23796T4+306650T5+3040264T6+20691976T7+3040264pT8+306650p2T9+23796p3T10+2642p4T11+188p5T12+6p6T13+p7T14 1 + 6 T + 188 T^{2} + 2642 T^{3} + 23796 T^{4} + 306650 T^{5} + 3040264 T^{6} + 20691976 T^{7} + 3040264 p T^{8} + 306650 p^{2} T^{9} + 23796 p^{3} T^{10} + 2642 p^{4} T^{11} + 188 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14}
71 1−39T+1033T2−19420T3+297470T4−3723376T5+39822981T6−360875937T7+39822981pT8−3723376p2T9+297470p3T10−19420p4T11+1033p5T12−39p6T13+p7T14 1 - 39 T + 1033 T^{2} - 19420 T^{3} + 297470 T^{4} - 3723376 T^{5} + 39822981 T^{6} - 360875937 T^{7} + 39822981 p T^{8} - 3723376 p^{2} T^{9} + 297470 p^{3} T^{10} - 19420 p^{4} T^{11} + 1033 p^{5} T^{12} - 39 p^{6} T^{13} + p^{7} T^{14}
73 1+4T+433T2+1378T3+84678T4+217302T5+9725129T6+20117620T7+9725129pT8+217302p2T9+84678p3T10+1378p4T11+433p5T12+4p6T13+p7T14 1 + 4 T + 433 T^{2} + 1378 T^{3} + 84678 T^{4} + 217302 T^{5} + 9725129 T^{6} + 20117620 T^{7} + 9725129 p T^{8} + 217302 p^{2} T^{9} + 84678 p^{3} T^{10} + 1378 p^{4} T^{11} + 433 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14}
79 1−21T+567T2−106pT3+133576T4−1497888T5+17422579T6−152791853T7+17422579pT8−1497888p2T9+133576p3T10−106p5T11+567p5T12−21p6T13+p7T14 1 - 21 T + 567 T^{2} - 106 p T^{3} + 133576 T^{4} - 1497888 T^{5} + 17422579 T^{6} - 152791853 T^{7} + 17422579 p T^{8} - 1497888 p^{2} T^{9} + 133576 p^{3} T^{10} - 106 p^{5} T^{11} + 567 p^{5} T^{12} - 21 p^{6} T^{13} + p^{7} T^{14}
83 1−15T+414T2−4607T3+68030T4−624005T5+6744800T6−57163121T7+6744800pT8−624005p2T9+68030p3T10−4607p4T11+414p5T12−15p6T13+p7T14 1 - 15 T + 414 T^{2} - 4607 T^{3} + 68030 T^{4} - 624005 T^{5} + 6744800 T^{6} - 57163121 T^{7} + 6744800 p T^{8} - 624005 p^{2} T^{9} + 68030 p^{3} T^{10} - 4607 p^{4} T^{11} + 414 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14}
89 1−13T+401T2−5034T3+85044T4−902320T5+11588685T6−98948575T7+11588685pT8−902320p2T9+85044p3T10−5034p4T11+401p5T12−13p6T13+p7T14 1 - 13 T + 401 T^{2} - 5034 T^{3} + 85044 T^{4} - 902320 T^{5} + 11588685 T^{6} - 98948575 T^{7} + 11588685 p T^{8} - 902320 p^{2} T^{9} + 85044 p^{3} T^{10} - 5034 p^{4} T^{11} + 401 p^{5} T^{12} - 13 p^{6} T^{13} + p^{7} T^{14}
97 1+4T+209T2+1362T3+35936T4+254438T5+4487773T6+27053728T7+4487773pT8+254438p2T9+35936p3T10+1362p4T11+209p5T12+4p6T13+p7T14 1 + 4 T + 209 T^{2} + 1362 T^{3} + 35936 T^{4} + 254438 T^{5} + 4487773 T^{6} + 27053728 T^{7} + 4487773 p T^{8} + 254438 p^{2} T^{9} + 35936 p^{3} T^{10} + 1362 p^{4} T^{11} + 209 p^{5} T^{12} + 4 p^{6} T^{13} + p^{7} T^{14}
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   L(s)=∏p ∏j=114(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.32423457717477854872680787337, −4.23772515332942656249178921838, −4.09361894890067103364869562717, −3.97373644405628991896590119856, −3.65929456737270177859443667004, −3.63802659008398983633233615074, −3.59478924863030054221177332663, −3.42387014926094513277114877021, −3.42374169207616093391104773936, −3.31671748581748399603682858719, −2.89803824266721195106099238268, −2.84785314088186182278795014423, −2.46821678307035284204214865241, −2.45539372791089766333096787716, −2.38183794725398940136413427448, −1.94095112426756435184080673079, −1.90717743731700716161288814707, −1.90107182878510573559145157483, −1.70683293323315154287678161540, −1.25048820984180224348414809615, −1.08336537325345904003292173584, −0.881111248182183080275837920457, −0.820000338745602561813494757279, −0.63940104413889996952447650411, −0.41359203633662299368586027131, 0.41359203633662299368586027131, 0.63940104413889996952447650411, 0.820000338745602561813494757279, 0.881111248182183080275837920457, 1.08336537325345904003292173584, 1.25048820984180224348414809615, 1.70683293323315154287678161540, 1.90107182878510573559145157483, 1.90717743731700716161288814707, 1.94095112426756435184080673079, 2.38183794725398940136413427448, 2.45539372791089766333096787716, 2.46821678307035284204214865241, 2.84785314088186182278795014423, 2.89803824266721195106099238268, 3.31671748581748399603682858719, 3.42374169207616093391104773936, 3.42387014926094513277114877021, 3.59478924863030054221177332663, 3.63802659008398983633233615074, 3.65929456737270177859443667004, 3.97373644405628991896590119856, 4.09361894890067103364869562717, 4.23772515332942656249178921838, 4.32423457717477854872680787337

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.