Properties

Label 12-1815e6-1.1-c1e6-0-2
Degree $12$
Conductor $3.575\times 10^{19}$
Sign $1$
Analytic cond. $9.26664\times 10^{6}$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 4-s + 6·5-s + 21·9-s − 6·12-s − 36·15-s − 4·16-s + 6·20-s − 4·23-s + 21·25-s − 56·27-s + 18·31-s + 21·36-s + 10·37-s + 126·45-s + 24·48-s + 13·49-s − 16·53-s − 20·59-s − 36·60-s − 5·64-s + 10·67-s + 24·69-s − 4·71-s − 126·75-s − 24·80-s + 126·81-s + ⋯
L(s)  = 1  − 3.46·3-s + 1/2·4-s + 2.68·5-s + 7·9-s − 1.73·12-s − 9.29·15-s − 16-s + 1.34·20-s − 0.834·23-s + 21/5·25-s − 10.7·27-s + 3.23·31-s + 7/2·36-s + 1.64·37-s + 18.7·45-s + 3.46·48-s + 13/7·49-s − 2.19·53-s − 2.60·59-s − 4.64·60-s − 5/8·64-s + 1.22·67-s + 2.88·69-s − 0.474·71-s − 14.5·75-s − 2.68·80-s + 14·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(3^{6} \cdot 5^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(9.26664\times 10^{6}\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 3^{6} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.475946831\)
\(L(\frac12)\) \(\approx\) \(4.475946831\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + T )^{6} \)
5 \( ( 1 - T )^{6} \)
11 \( 1 \)
good2 \( 1 - T^{2} + 5 T^{4} - p^{2} T^{6} + 5 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \)
7 \( 1 - 13 T^{2} + 183 T^{4} - 1286 T^{6} + 183 p^{2} T^{8} - 13 p^{4} T^{10} + p^{6} T^{12} \)
13 \( 1 + 2 p T^{2} + 615 T^{4} + 8524 T^{6} + 615 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \)
17 \( 1 + 20 T^{2} + 824 T^{4} + 9854 T^{6} + 824 p^{2} T^{8} + 20 p^{4} T^{10} + p^{6} T^{12} \)
19 \( 1 - T^{2} + 123 T^{4} + 94 p T^{6} + 123 p^{2} T^{8} - p^{4} T^{10} + p^{6} T^{12} \)
23 \( ( 1 + 2 T + 40 T^{2} + 26 T^{3} + 40 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
29 \( 1 + 122 T^{2} + 7367 T^{4} + 267788 T^{6} + 7367 p^{2} T^{8} + 122 p^{4} T^{10} + p^{6} T^{12} \)
31 \( ( 1 - 9 T + 48 T^{2} - 7 p T^{3} + 48 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 - 5 T + 89 T^{2} - 278 T^{3} + 89 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{3} \)
43 \( 1 + 66 T^{2} + 1623 T^{4} + 38972 T^{6} + 1623 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} \)
47 \( ( 1 + 72 T^{2} + 216 T^{3} + 72 p T^{4} + p^{3} T^{6} )^{2} \)
53 \( ( 1 + 8 T + 160 T^{2} + 794 T^{3} + 160 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
59 \( ( 1 + 10 T + 61 T^{2} + 268 T^{3} + 61 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( 1 + 125 T^{2} + 12702 T^{4} + 764233 T^{6} + 12702 p^{2} T^{8} + 125 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 - 5 T + 179 T^{2} - 578 T^{3} + 179 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
71 \( ( 1 + 2 T + 133 T^{2} + 380 T^{3} + 133 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 + 291 T^{2} + 42627 T^{4} + 3837890 T^{6} + 42627 p^{2} T^{8} + 291 p^{4} T^{10} + p^{6} T^{12} \)
79 \( 1 + 65 T^{2} + 6462 T^{4} + 16357 T^{6} + 6462 p^{2} T^{8} + 65 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 + 314 T^{2} + 50711 T^{4} + 5124716 T^{6} + 50711 p^{2} T^{8} + 314 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 24 T + 387 T^{2} - 4056 T^{3} + 387 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( ( 1 - T + 65 T^{2} + 650 T^{3} + 65 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.95973412027326675790043206767, −4.67403459658097582716906542838, −4.60126583134952915227422915190, −4.42336546718762866628302272030, −4.36371412397755383991477360608, −4.35581702255911972604203468346, −4.29088769115292869786994751798, −3.76898140057918541120645541967, −3.61499695995371594337187527010, −3.35530872173415032559021292196, −3.30495458457395109417196497002, −2.97967487707175733839600524765, −2.89453091026976250854044266212, −2.62591865621093436354674313949, −2.33039536642846141435141295528, −2.32531104409938818693527053169, −1.96760625177718074842108364427, −1.96169091183883981829829126970, −1.86176657611354719519191783647, −1.39463534143334385091833402099, −1.31072668184384308454318721426, −1.03743042228864563946099827554, −0.803600380337510547049285616570, −0.53106185488785757090461552161, −0.42237175358582420595043809135, 0.42237175358582420595043809135, 0.53106185488785757090461552161, 0.803600380337510547049285616570, 1.03743042228864563946099827554, 1.31072668184384308454318721426, 1.39463534143334385091833402099, 1.86176657611354719519191783647, 1.96169091183883981829829126970, 1.96760625177718074842108364427, 2.32531104409938818693527053169, 2.33039536642846141435141295528, 2.62591865621093436354674313949, 2.89453091026976250854044266212, 2.97967487707175733839600524765, 3.30495458457395109417196497002, 3.35530872173415032559021292196, 3.61499695995371594337187527010, 3.76898140057918541120645541967, 4.29088769115292869786994751798, 4.35581702255911972604203468346, 4.36371412397755383991477360608, 4.42336546718762866628302272030, 4.60126583134952915227422915190, 4.67403459658097582716906542838, 4.95973412027326675790043206767

Graph of the $Z$-function along the critical line

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