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Properties

Label 12-357e6-1.1-c1e6-0-0
Degree 1212
Conductor 2.070×10152.070\times 10^{15}
Sign 11
Analytic cond. 536.626536.626
Root an. cond. 1.688381.68838
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  + 4·2-s + 4·4-s − 4·8-s − 3·9-s − 2·13-s − 8·16-s − 12·17-s − 12·18-s + 14·19-s + 19·25-s − 8·26-s − 48·34-s − 12·36-s + 56·38-s + 2·43-s + 28·47-s − 3·49-s + 76·50-s − 8·52-s − 12·53-s + 28·59-s − 4·64-s − 24·67-s − 48·68-s + 12·72-s + 56·76-s + 6·81-s + ⋯
L(s)  = 1  + 2.82·2-s + 2·4-s − 1.41·8-s − 9-s − 0.554·13-s − 2·16-s − 2.91·17-s − 2.82·18-s + 3.21·19-s + 19/5·25-s − 1.56·26-s − 8.23·34-s − 2·36-s + 9.08·38-s + 0.304·43-s + 4.08·47-s − 3/7·49-s + 10.7·50-s − 1.10·52-s − 1.64·53-s + 3.64·59-s − 1/2·64-s − 2.93·67-s − 5.82·68-s + 1.41·72-s + 6.42·76-s + 2/3·81-s + ⋯

Functional equation

Λ(s)=((36⋅76⋅176)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((36⋅76⋅176)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 36⋅76⋅1763^{6} \cdot 7^{6} \cdot 17^{6}
Sign: 11
Analytic conductor: 536.626536.626
Root analytic conductor: 1.688381.68838
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 36⋅76⋅176, ( :[1/2]6), 1)(12,\ 3^{6} \cdot 7^{6} \cdot 17^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 7.8904516617.890451661
L(12)L(\frac12) ≈\approx 7.8904516617.890451661
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+T2)3 ( 1 + T^{2} )^{3}
7 (1+T2)3 ( 1 + T^{2} )^{3}
17 1+12T+83T2+392T3+83pT4+12p2T5+p3T6 1 + 12 T + 83 T^{2} + 392 T^{3} + 83 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
good2 (1−pT+p2T2−3pT3+p3T4−p3T5+p3T6)2 ( 1 - p T + p^{2} T^{2} - 3 p T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} )^{2}
5 1−19T2+162T4−919T6+162p2T8−19p4T10+p6T12 1 - 19 T^{2} + 162 T^{4} - 919 T^{6} + 162 p^{2} T^{8} - 19 p^{4} T^{10} + p^{6} T^{12}
11 1−47T2+1038T4−14099T6+1038p2T8−47p4T10+p6T12 1 - 47 T^{2} + 1038 T^{4} - 14099 T^{6} + 1038 p^{2} T^{8} - 47 p^{4} T^{10} + p^{6} T^{12}
13 (1+T+2pT2+3T3+2p2T4+p2T5+p3T6)2 ( 1 + T + 2 p T^{2} + 3 T^{3} + 2 p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2}
19 (1−7T+70T2−271T3+70pT4−7p2T5+p3T6)2 ( 1 - 7 T + 70 T^{2} - 271 T^{3} + 70 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2}
23 1−71T2+2478T4−63203T6+2478p2T8−71p4T10+p6T12 1 - 71 T^{2} + 2478 T^{4} - 63203 T^{6} + 2478 p^{2} T^{8} - 71 p^{4} T^{10} + p^{6} T^{12}
29 1−162T2+11255T4−429068T6+11255p2T8−162p4T10+p6T12 1 - 162 T^{2} + 11255 T^{4} - 429068 T^{6} + 11255 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12}
31 1−158T2+11047T4−440720T6+11047p2T8−158p4T10+p6T12 1 - 158 T^{2} + 11047 T^{4} - 440720 T^{6} + 11047 p^{2} T^{8} - 158 p^{4} T^{10} + p^{6} T^{12}
37 1−154T2+10891T4−484912T6+10891p2T8−154p4T10+p6T12 1 - 154 T^{2} + 10891 T^{4} - 484912 T^{6} + 10891 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12}
41 1−223T2+21542T4−1154535T6+21542p2T8−223p4T10+p6T12 1 - 223 T^{2} + 21542 T^{4} - 1154535 T^{6} + 21542 p^{2} T^{8} - 223 p^{4} T^{10} + p^{6} T^{12}
43 (1−T+76T2−217T3+76pT4−p2T5+p3T6)2 ( 1 - T + 76 T^{2} - 217 T^{3} + 76 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2}
47 (1−14T+145T2−1014T3+145pT4−14p2T5+p3T6)2 ( 1 - 14 T + 145 T^{2} - 1014 T^{3} + 145 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2}
53 (1+6T+137T2+514T3+137pT4+6p2T5+p3T6)2 ( 1 + 6 T + 137 T^{2} + 514 T^{3} + 137 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
59 (1−14T+193T2−1422T3+193pT4−14p2T5+p3T6)2 ( 1 - 14 T + 193 T^{2} - 1422 T^{3} + 193 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2}
61 1−82T2+1775T4−60168T6+1775p2T8−82p4T10+p6T12 1 - 82 T^{2} + 1775 T^{4} - 60168 T^{6} + 1775 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12}
67 (1+12T+89T2+604T3+89pT4+12p2T5+p3T6)2 ( 1 + 12 T + 89 T^{2} + 604 T^{3} + 89 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}
71 1−98T2+11727T4−845420T6+11727p2T8−98p4T10+p6T12 1 - 98 T^{2} + 11727 T^{4} - 845420 T^{6} + 11727 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12}
73 1−138T2+18303T4−1472524T6+18303p2T8−138p4T10+p6T12 1 - 138 T^{2} + 18303 T^{4} - 1472524 T^{6} + 18303 p^{2} T^{8} - 138 p^{4} T^{10} + p^{6} T^{12}
79 1+46T2+715T4−199424T6+715p2T8+46p4T10+p6T12 1 + 46 T^{2} + 715 T^{4} - 199424 T^{6} + 715 p^{2} T^{8} + 46 p^{4} T^{10} + p^{6} T^{12}
83 (1−16T+273T2−2640T3+273pT4−16p2T5+p3T6)2 ( 1 - 16 T + 273 T^{2} - 2640 T^{3} + 273 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2}
89 (1+8T+203T2+1504T3+203pT4+8p2T5+p3T6)2 ( 1 + 8 T + 203 T^{2} + 1504 T^{3} + 203 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
97 1−250T2+43535T4−4887660T6+43535p2T8−250p4T10+p6T12 1 - 250 T^{2} + 43535 T^{4} - 4887660 T^{6} + 43535 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.94198037855641249373282410753, −5.84528103235900689940143389184, −5.65372129823184107658826654602, −5.60295335655990068079615962212, −5.41503879342149395752606968810, −5.22607051775167748358954933808, −5.09582969945548044977539813874, −4.75441413740991324765730204411, −4.62507215911845802263415741882, −4.53311752144200065891665574188, −4.46744778039192477458673128981, −4.36768887542244328533530474927, −4.26399080512577102550800242519, −3.58215024939891053160806191324, −3.51853465519705374927020909830, −3.43226185131747918955277691799, −3.22529083826253773089832408657, −3.13843481398489898228176580190, −2.60927948028672519806040595415, −2.46596390334713804129975898683, −2.28261966680719741529164725766, −2.04241830571489699337641536563, −1.30623489778382231548444434294, −0.879783169499310090275501419239, −0.65330085683607734884152749976, 0.65330085683607734884152749976, 0.879783169499310090275501419239, 1.30623489778382231548444434294, 2.04241830571489699337641536563, 2.28261966680719741529164725766, 2.46596390334713804129975898683, 2.60927948028672519806040595415, 3.13843481398489898228176580190, 3.22529083826253773089832408657, 3.43226185131747918955277691799, 3.51853465519705374927020909830, 3.58215024939891053160806191324, 4.26399080512577102550800242519, 4.36768887542244328533530474927, 4.46744778039192477458673128981, 4.53311752144200065891665574188, 4.62507215911845802263415741882, 4.75441413740991324765730204411, 5.09582969945548044977539813874, 5.22607051775167748358954933808, 5.41503879342149395752606968810, 5.60295335655990068079615962212, 5.65372129823184107658826654602, 5.84528103235900689940143389184, 5.94198037855641249373282410753

Graph of the ZZ-function along the critical line

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