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Properties

Label 16-1110e8-1.1-c1e8-0-1
Degree 1616
Conductor 2.305×10242.305\times 10^{24}
Sign 11
Analytic cond. 3.80890×1073.80890\times 10^{7}
Root an. cond. 2.977142.97714
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  + 8·3-s − 4·4-s − 4·7-s + 36·9-s − 4·11-s − 32·12-s + 10·16-s − 32·21-s − 4·25-s + 120·27-s + 16·28-s − 32·33-s − 144·36-s + 18·37-s − 10·41-s + 16·44-s + 28·47-s + 80·48-s − 6·49-s + 12·53-s − 144·63-s − 20·64-s + 12·67-s + 24·71-s + 10·73-s − 32·75-s + 16·77-s + ⋯
L(s)  = 1  + 4.61·3-s − 2·4-s − 1.51·7-s + 12·9-s − 1.20·11-s − 9.23·12-s + 5/2·16-s − 6.98·21-s − 4/5·25-s + 23.0·27-s + 3.02·28-s − 5.57·33-s − 24·36-s + 2.95·37-s − 1.56·41-s + 2.41·44-s + 4.08·47-s + 11.5·48-s − 6/7·49-s + 1.64·53-s − 18.1·63-s − 5/2·64-s + 1.46·67-s + 2.84·71-s + 1.17·73-s − 3.69·75-s + 1.82·77-s + ⋯

Functional equation

Λ(s)=((28⋅38⋅58⋅378)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((28⋅38⋅58⋅378)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 28⋅38⋅58⋅3782^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8}
Sign: 11
Analytic conductor: 3.80890×1073.80890\times 10^{7}
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 28⋅38⋅58⋅378, ( :[1/2]8), 1)(16,\ 2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 37^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 23.1534740223.15347402
L(12)L(\frac12) ≈\approx 23.1534740223.15347402
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)
bad2 (1+T2)4 ( 1 + T^{2} )^{4}
3 (1−T)8 ( 1 - T )^{8}
5 (1+T2)4 ( 1 + T^{2} )^{4}
37 1−18T+120T2−230T3−610T4−230pT5+120p2T6−18p3T7+p4T8 1 - 18 T + 120 T^{2} - 230 T^{3} - 610 T^{4} - 230 p T^{5} + 120 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
good7 (1+2T+9T2+6T3+44T4+6pT5+9p2T6+2p3T7+p4T8)2 ( 1 + 2 T + 9 T^{2} + 6 T^{3} + 44 T^{4} + 6 p T^{5} + 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
11 (1+2T+pT2+28T3+60T4+28pT5+p3T6+2p3T7+p4T8)2 ( 1 + 2 T + p T^{2} + 28 T^{3} + 60 T^{4} + 28 p T^{5} + p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}
13 1−51T2+1450T4−27925T6+412138T8−27925p2T10+1450p4T12−51p6T14+p8T16 1 - 51 T^{2} + 1450 T^{4} - 27925 T^{6} + 412138 T^{8} - 27925 p^{2} T^{10} + 1450 p^{4} T^{12} - 51 p^{6} T^{14} + p^{8} T^{16}
17 1−86T2+3449T4−88262T6+1685524T8−88262p2T10+3449p4T12−86p6T14+p8T16 1 - 86 T^{2} + 3449 T^{4} - 88262 T^{6} + 1685524 T^{8} - 88262 p^{2} T^{10} + 3449 p^{4} T^{12} - 86 p^{6} T^{14} + p^{8} T^{16}
19 1−71T2+3066T4−90321T6+1978538T8−90321p2T10+3066p4T12−71p6T14+p8T16 1 - 71 T^{2} + 3066 T^{4} - 90321 T^{6} + 1978538 T^{8} - 90321 p^{2} T^{10} + 3066 p^{4} T^{12} - 71 p^{6} T^{14} + p^{8} T^{16}
23 1−131T2+8350T4−334125T6+9172738T8−334125p2T10+8350p4T12−131p6T14+p8T16 1 - 131 T^{2} + 8350 T^{4} - 334125 T^{6} + 9172738 T^{8} - 334125 p^{2} T^{10} + 8350 p^{4} T^{12} - 131 p^{6} T^{14} + p^{8} T^{16}
29 1−19T2+1426T4−9669T6+1269306T8−9669p2T10+1426p4T12−19p6T14+p8T16 1 - 19 T^{2} + 1426 T^{4} - 9669 T^{6} + 1269306 T^{8} - 9669 p^{2} T^{10} + 1426 p^{4} T^{12} - 19 p^{6} T^{14} + p^{8} T^{16}
31 1−131T2+8502T4−362061T6+12194738T8−362061p2T10+8502p4T12−131p6T14+p8T16 1 - 131 T^{2} + 8502 T^{4} - 362061 T^{6} + 12194738 T^{8} - 362061 p^{2} T^{10} + 8502 p^{4} T^{12} - 131 p^{6} T^{14} + p^{8} T^{16}
41 (1+5T+118T2+563T3+6674T4+563pT5+118p2T6+5p3T7+p4T8)2 ( 1 + 5 T + 118 T^{2} + 563 T^{3} + 6674 T^{4} + 563 p T^{5} + 118 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2}
43 1−255T2+30706T4−2298169T6+117805018T8−2298169p2T10+30706p4T12−255p6T14+p8T16 1 - 255 T^{2} + 30706 T^{4} - 2298169 T^{6} + 117805018 T^{8} - 2298169 p^{2} T^{10} + 30706 p^{4} T^{12} - 255 p^{6} T^{14} + p^{8} T^{16}
47 (1−14T+64T2+586T3−7618T4+586pT5+64p2T6−14p3T7+p4T8)2 ( 1 - 14 T + 64 T^{2} + 586 T^{3} - 7618 T^{4} + 586 p T^{5} + 64 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2}
53 (1−6T+205T2−922T3+16124T4−922pT5+205p2T6−6p3T7+p4T8)2 ( 1 - 6 T + 205 T^{2} - 922 T^{3} + 16124 T^{4} - 922 p T^{5} + 205 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
59 1−260T2+36052T4−3325212T6+226302646T8−3325212p2T10+36052p4T12−260p6T14+p8T16 1 - 260 T^{2} + 36052 T^{4} - 3325212 T^{6} + 226302646 T^{8} - 3325212 p^{2} T^{10} + 36052 p^{4} T^{12} - 260 p^{6} T^{14} + p^{8} T^{16}
61 1−311T2+47598T4−77205pT6+334310210T8−77205p3T10+47598p4T12−311p6T14+p8T16 1 - 311 T^{2} + 47598 T^{4} - 77205 p T^{6} + 334310210 T^{8} - 77205 p^{3} T^{10} + 47598 p^{4} T^{12} - 311 p^{6} T^{14} + p^{8} T^{16}
67 (1−6T+60T2+74T3+3158T4+74pT5+60p2T6−6p3T7+p4T8)2 ( 1 - 6 T + 60 T^{2} + 74 T^{3} + 3158 T^{4} + 74 p T^{5} + 60 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
71 (1−12T+256T2−2300T3+26462T4−2300pT5+256p2T6−12p3T7+p4T8)2 ( 1 - 12 T + 256 T^{2} - 2300 T^{3} + 26462 T^{4} - 2300 p T^{5} + 256 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2}
73 (1−5T+210T2−867T3+21882T4−867pT5+210p2T6−5p3T7+p4T8)2 ( 1 - 5 T + 210 T^{2} - 867 T^{3} + 21882 T^{4} - 867 p T^{5} + 210 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2}
79 1−140T2+21204T4−2466036T6+185705942T8−2466036p2T10+21204p4T12−140p6T14+p8T16 1 - 140 T^{2} + 21204 T^{4} - 2466036 T^{6} + 185705942 T^{8} - 2466036 p^{2} T^{10} + 21204 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16}
83 (1−3T+208T2+5T3+20174T4+5pT5+208p2T6−3p3T7+p4T8)2 ( 1 - 3 T + 208 T^{2} + 5 T^{3} + 20174 T^{4} + 5 p T^{5} + 208 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2}
89 1−527T2+132910T4−20902113T6+2232605602T8−20902113p2T10+132910p4T12−527p6T14+p8T16 1 - 527 T^{2} + 132910 T^{4} - 20902113 T^{6} + 2232605602 T^{8} - 20902113 p^{2} T^{10} + 132910 p^{4} T^{12} - 527 p^{6} T^{14} + p^{8} T^{16}
97 1−275T2+62634T4−8629437T6+1011478730T8−8629437p2T10+62634p4T12−275p6T14+p8T16 1 - 275 T^{2} + 62634 T^{4} - 8629437 T^{6} + 1011478730 T^{8} - 8629437 p^{2} T^{10} + 62634 p^{4} T^{12} - 275 p^{6} T^{14} + p^{8} T^{16}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.04746385607230922513384260244, −4.02547163112643252288104562762, −3.92330080889104943902022518475, −3.78038841065032231900085266698, −3.61934492578378904858909251691, −3.54711612946183544983508582897, −3.51886649165010669220867172732, −3.37390782374923872688219762636, −3.18997448682146557318369095360, −2.93937626845352026859992847508, −2.74263697392464393422584078254, −2.61108377119400778804633141053, −2.60696791974846211043663761359, −2.48200156597610685879991356126, −2.45048548525437259149616380828, −2.40818767875137690903752965027, −2.25555573660070458621494075485, −1.75279153798382622683326816128, −1.64349430186259516883569783444, −1.40684933185677598087008931822, −1.35741105328725901855168827261, −1.18847311637681584561643726167, −0.71282682693049078513423651912, −0.48672693371614329752513174304, −0.43816467838509020214220256905, 0.43816467838509020214220256905, 0.48672693371614329752513174304, 0.71282682693049078513423651912, 1.18847311637681584561643726167, 1.35741105328725901855168827261, 1.40684933185677598087008931822, 1.64349430186259516883569783444, 1.75279153798382622683326816128, 2.25555573660070458621494075485, 2.40818767875137690903752965027, 2.45048548525437259149616380828, 2.48200156597610685879991356126, 2.60696791974846211043663761359, 2.61108377119400778804633141053, 2.74263697392464393422584078254, 2.93937626845352026859992847508, 3.18997448682146557318369095360, 3.37390782374923872688219762636, 3.51886649165010669220867172732, 3.54711612946183544983508582897, 3.61934492578378904858909251691, 3.78038841065032231900085266698, 3.92330080889104943902022518475, 4.02547163112643252288104562762, 4.04746385607230922513384260244

Graph of the ZZ-function along the critical line

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