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Properties

Label 12-378e6-1.1-c1e6-0-1
Degree 1212
Conductor 2.917×10152.917\times 10^{15}
Sign 11
Analytic cond. 756.159756.159
Root an. cond. 1.737331.73733
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·4-s + 2·5-s − 4·7-s + 2·8-s − 6·10-s − 2·11-s + 8·13-s + 12·14-s − 9·16-s + 4·17-s − 3·19-s + 6·20-s + 6·22-s − 14·23-s − 15·25-s − 24·26-s − 12·28-s + 5·29-s + 20·31-s + 9·32-s − 12·34-s − 8·35-s + 3·37-s + 9·38-s + 4·40-s − 6·43-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s + 0.894·5-s − 1.51·7-s + 0.707·8-s − 1.89·10-s − 0.603·11-s + 2.21·13-s + 3.20·14-s − 9/4·16-s + 0.970·17-s − 0.688·19-s + 1.34·20-s + 1.27·22-s − 2.91·23-s − 3·25-s − 4.70·26-s − 2.26·28-s + 0.928·29-s + 3.59·31-s + 1.59·32-s − 2.05·34-s − 1.35·35-s + 0.493·37-s + 1.45·38-s + 0.632·40-s − 0.914·43-s + ⋯

Functional equation

Λ(s)=((26⋅318⋅76)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((26⋅318⋅76)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{18} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 26⋅318⋅762^{6} \cdot 3^{18} \cdot 7^{6}
Sign: 11
Analytic conductor: 756.159756.159
Root analytic conductor: 1.737331.73733
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 26⋅318⋅76, ( :[1/2]6), 1)(12,\ 2^{6} \cdot 3^{18} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.54608668190.5460866819
L(12)L(\frac12) ≈\approx 0.54608668190.5460866819
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+T+T2)3 ( 1 + T + T^{2} )^{3}
3 1 1
7 1+4T+2pT2+55T3+2p2T4+4p2T5+p3T6 1 + 4 T + 2 p T^{2} + 55 T^{3} + 2 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
good5 (1−T+9T2−13T3+9pT4−p2T5+p3T6)2 ( 1 - T + 9 T^{2} - 13 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2}
11 (1+T+27T2+25T3+27pT4+p2T5+p3T6)2 ( 1 + T + 27 T^{2} + 25 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2}
13 1−8T+24T2−42T3−32T4+1408T5−7901T6+1408pT7−32p2T8−42p3T9+24p4T10−8p5T11+p6T12 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 1408 p T^{7} - 32 p^{2} T^{8} - 42 p^{3} T^{9} + 24 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
17 1−4T−23T2+4pT3+410T4−220T5−8111T6−220pT7+410p2T8+4p4T9−23p4T10−4p5T11+p6T12 1 - 4 T - 23 T^{2} + 4 p T^{3} + 410 T^{4} - 220 T^{5} - 8111 T^{6} - 220 p T^{7} + 410 p^{2} T^{8} + 4 p^{4} T^{9} - 23 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
19 1+3T−12T2−67T3−153T4+54T5+6315T6+54pT7−153p2T8−67p3T9−12p4T10+3p5T11+p6T12 1 + 3 T - 12 T^{2} - 67 T^{3} - 153 T^{4} + 54 T^{5} + 6315 T^{6} + 54 p T^{7} - 153 p^{2} T^{8} - 67 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
23 (1+7T+81T2+325T3+81pT4+7p2T5+p3T6)2 ( 1 + 7 T + 81 T^{2} + 325 T^{3} + 81 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2}
29 1−5T+4T2−251T3+197T4+3418T5+20293T6+3418pT7+197p2T8−251p3T9+4p4T10−5p5T11+p6T12 1 - 5 T + 4 T^{2} - 251 T^{3} + 197 T^{4} + 3418 T^{5} + 20293 T^{6} + 3418 p T^{7} + 197 p^{2} T^{8} - 251 p^{3} T^{9} + 4 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12}
31 1−20T+6pT2−1398T3+10342T4−62234T5+331987T6−62234pT7+10342p2T8−1398p3T9+6p5T10−20p5T11+p6T12 1 - 20 T + 6 p T^{2} - 1398 T^{3} + 10342 T^{4} - 62234 T^{5} + 331987 T^{6} - 62234 p T^{7} + 10342 p^{2} T^{8} - 1398 p^{3} T^{9} + 6 p^{5} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12}
37 (1−11T+pT2)3(1+10T+pT2)3 ( 1 - 11 T + p T^{2} )^{3}( 1 + 10 T + p T^{2} )^{3}
41 1−90T2−18T3+4410T4+810T5−194177T6+810pT7+4410p2T8−18p3T9−90p4T10+p6T12 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 810 p T^{7} + 4410 p^{2} T^{8} - 18 p^{3} T^{9} - 90 p^{4} T^{10} + p^{6} T^{12}
43 (1−12T−6T2+547T3−6pT4−12p2T5+p3T6)(1+18T+198T2+1519T3+198pT4+18p2T5+p3T6) ( 1 - 12 T - 6 T^{2} + 547 T^{3} - 6 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )( 1 + 18 T + 198 T^{2} + 1519 T^{3} + 198 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )
47 1−9T−6T2+531T3−2433T4−3438T5+104623T6−3438pT7−2433p2T8+531p3T9−6p4T10−9p5T11+p6T12 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 3438 p T^{7} - 2433 p^{2} T^{8} + 531 p^{3} T^{9} - 6 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
53 1+15T+33T3+13635T4+60360T5−225155T6+60360pT7+13635p2T8+33p3T9+15p5T11+p6T12 1 + 15 T + 33 T^{3} + 13635 T^{4} + 60360 T^{5} - 225155 T^{6} + 60360 p T^{7} + 13635 p^{2} T^{8} + 33 p^{3} T^{9} + 15 p^{5} T^{11} + p^{6} T^{12}
59 1−14T−20T2+154T3+11666T4−35126T5−499301T6−35126pT7+11666p2T8+154p3T9−20p4T10−14p5T11+p6T12 1 - 14 T - 20 T^{2} + 154 T^{3} + 11666 T^{4} - 35126 T^{5} - 499301 T^{6} - 35126 p T^{7} + 11666 p^{2} T^{8} + 154 p^{3} T^{9} - 20 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12}
61 1−8T−114T2+342T3+13762T4−13214T5−937217T6−13214pT7+13762p2T8+342p3T9−114p4T10−8p5T11+p6T12 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 13214 p T^{7} + 13762 p^{2} T^{8} + 342 p^{3} T^{9} - 114 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
67 1−T−88T2−243T3+2035T4+14290T5+72259T6+14290pT7+2035p2T8−243p3T9−88p4T10−p5T11+p6T12 1 - T - 88 T^{2} - 243 T^{3} + 2035 T^{4} + 14290 T^{5} + 72259 T^{6} + 14290 p T^{7} + 2035 p^{2} T^{8} - 243 p^{3} T^{9} - 88 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12}
71 (1+7T+15T2−599T3+15pT4+7p2T5+p3T6)2 ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 15 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1−19T+134T2−27T3−5759T4+41986T5−314903T6+41986pT7−5759p2T8−27p3T9+134p4T10−19p5T11+p6T12 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 41986 p T^{7} - 5759 p^{2} T^{8} - 27 p^{3} T^{9} + 134 p^{4} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12}
79 1−5T−138T2+123T3+11347T4+21118T5−1048937T6+21118pT7+11347p2T8+123p3T9−138p4T10−5p5T11+p6T12 1 - 5 T - 138 T^{2} + 123 T^{3} + 11347 T^{4} + 21118 T^{5} - 1048937 T^{6} + 21118 p T^{7} + 11347 p^{2} T^{8} + 123 p^{3} T^{9} - 138 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12}
83 1+2T−182T2+2T3+18788T4−13564T5−1721225T6−13564pT7+18788p2T8+2p3T9−182p4T10+2p5T11+p6T12 1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 13564 p T^{7} + 18788 p^{2} T^{8} + 2 p^{3} T^{9} - 182 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
89 1−9T−144T2+1197T3+16101T4−73314T5−1141967T6−73314pT7+16101p2T8+1197p3T9−144p4T10−9p5T11+p6T12 1 - 9 T - 144 T^{2} + 1197 T^{3} + 16101 T^{4} - 73314 T^{5} - 1141967 T^{6} - 73314 p T^{7} + 16101 p^{2} T^{8} + 1197 p^{3} T^{9} - 144 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
97 1−28T+281T2−2724T3+45178T4−388196T5+2169217T6−388196pT7+45178p2T8−2724p3T9+281p4T10−28p5T11+p6T12 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 388196 p T^{7} + 45178 p^{2} T^{8} - 2724 p^{3} T^{9} + 281 p^{4} T^{10} - 28 p^{5} T^{11} + 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

−6.33494758858488591759251296799, −6.02304779982039994760018645355, −5.89381929066827786350808604127, −5.67587743052420171466070092200, −5.65171965761350440612036963254, −5.55940084735348107065927736712, −5.06043877812093635130566172818, −4.81664585564264533858711100717, −4.81146958152562143543122396892, −4.35302998245710172781996760259, −4.19174952634560981867959741388, −3.99317401508695538031094859079, −3.86472451637733685311821579134, −3.69721634376078247122664610306, −3.60050624706311344299279063644, −2.99197892063502348043752662188, −2.97002455780179579234501584556, −2.58332120179096718550293936343, −2.55147765920002646137517570120, −1.92734219114311631730865400329, −1.80486715827586510244283793887, −1.72543276160858166446234379667, −1.21296200663693563952143453751, −0.64848224129327926610869568510, −0.47917294819184509109412475082, 0.47917294819184509109412475082, 0.64848224129327926610869568510, 1.21296200663693563952143453751, 1.72543276160858166446234379667, 1.80486715827586510244283793887, 1.92734219114311631730865400329, 2.55147765920002646137517570120, 2.58332120179096718550293936343, 2.97002455780179579234501584556, 2.99197892063502348043752662188, 3.60050624706311344299279063644, 3.69721634376078247122664610306, 3.86472451637733685311821579134, 3.99317401508695538031094859079, 4.19174952634560981867959741388, 4.35302998245710172781996760259, 4.81146958152562143543122396892, 4.81664585564264533858711100717, 5.06043877812093635130566172818, 5.55940084735348107065927736712, 5.65171965761350440612036963254, 5.67587743052420171466070092200, 5.89381929066827786350808604127, 6.02304779982039994760018645355, 6.33494758858488591759251296799

Graph of the ZZ-function along the critical line

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