Properties

Label 12-1680e6-1.1-c1e6-0-1
Degree $12$
Conductor $2.248\times 10^{19}$
Sign $1$
Analytic cond. $5.82798\times 10^{6}$
Root an. cond. $3.66263$
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  − 2·5-s − 3·9-s + 4·11-s − 4·19-s + 25-s + 12·29-s + 28·31-s − 20·41-s + 6·45-s − 3·49-s − 8·55-s − 16·59-s + 4·61-s + 36·71-s + 8·79-s + 6·81-s + 12·89-s + 8·95-s − 12·99-s − 4·101-s − 44·109-s + 18·121-s − 8·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.894·5-s − 9-s + 1.20·11-s − 0.917·19-s + 1/5·25-s + 2.22·29-s + 5.02·31-s − 3.12·41-s + 0.894·45-s − 3/7·49-s − 1.07·55-s − 2.08·59-s + 0.512·61-s + 4.27·71-s + 0.900·79-s + 2/3·81-s + 1.27·89-s + 0.820·95-s − 1.20·99-s − 0.398·101-s − 4.21·109-s + 1.63·121-s − 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(5.82798\times 10^{6}\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{24} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.312883337\)
\(L(\frac12)\) \(\approx\) \(2.312883337\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{2} )^{3} \)
5 \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7 \( ( 1 + T^{2} )^{3} \)
good11 \( ( 1 - 2 T - 3 T^{2} + 60 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
13 \( 1 - 18 T^{2} - 25 T^{4} + 2404 T^{6} - 25 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 26 T^{2} + 1007 T^{4} + 876 p T^{6} + 1007 p^{2} T^{8} + 26 p^{4} T^{10} + p^{6} T^{12} \)
19 \( ( 1 + 2 T - 3 T^{2} - 124 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
23 \( 1 + 6 T^{2} + 1151 T^{4} + 8788 T^{6} + 1151 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 2 T + p T^{2} )^{6} \)
31 \( ( 1 - 14 T + 145 T^{2} - 908 T^{3} + 145 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( 1 - 46 T^{2} + 2423 T^{4} - 52260 T^{6} + 2423 p^{2} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} \)
41 \( ( 1 + 10 T + 143 T^{2} + 812 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( 1 - 146 T^{2} + 11287 T^{4} - 585692 T^{6} + 11287 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \)
47 \( ( 1 - p T^{2} )^{6} \)
53 \( 1 - 226 T^{2} + 24215 T^{4} - 1593276 T^{6} + 24215 p^{2} T^{8} - 226 p^{4} T^{10} + p^{6} T^{12} \)
59 \( ( 1 + 8 T + 113 T^{2} + 688 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
61 \( ( 1 - 2 T - 29 T^{2} - 140 T^{3} - 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
67 \( 1 - 146 T^{2} + 6919 T^{4} - 152348 T^{6} + 6919 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \)
71 \( ( 1 - 18 T + 161 T^{2} - 1204 T^{3} + 161 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( 1 - 154 T^{2} + 2431 T^{4} + 519188 T^{6} + 2431 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \)
79 \( ( 1 - 4 T + 189 T^{2} - 568 T^{3} + 189 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \)
89 \( ( 1 - 6 T + 143 T^{2} - 1300 T^{3} + 143 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 458 T^{2} + 97423 T^{4} - 12066764 T^{6} + 97423 p^{2} T^{8} - 458 p^{4} T^{10} + p^{6} T^{12} \)
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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

−4.94900608211865650205103718124, −4.61897462494894681333250983222, −4.53228677761958922868746905640, −4.43686246709029978677315629674, −4.33017784552889387201559369052, −4.13077921822599670970705716554, −4.06470651024197752270995687351, −3.82582147463208900487057109343, −3.56202432856041743610982345722, −3.38129691002454990953104965376, −3.30987063563116237682606979886, −3.18745272688949554926810339142, −2.86349811258786289621682911177, −2.84820395713762471101419538693, −2.72872323708916814607659445729, −2.37971119372962361395838988519, −2.32418339933843958214773565752, −2.00503786719265638891576340206, −1.72951325100819977894398342628, −1.68771914612339091273721285716, −1.27109624379753640853342390219, −0.955003904261396408881979855788, −0.847910724087556684829637899944, −0.67470077047961518219938690437, −0.21429298850769099316222447182, 0.21429298850769099316222447182, 0.67470077047961518219938690437, 0.847910724087556684829637899944, 0.955003904261396408881979855788, 1.27109624379753640853342390219, 1.68771914612339091273721285716, 1.72951325100819977894398342628, 2.00503786719265638891576340206, 2.32418339933843958214773565752, 2.37971119372962361395838988519, 2.72872323708916814607659445729, 2.84820395713762471101419538693, 2.86349811258786289621682911177, 3.18745272688949554926810339142, 3.30987063563116237682606979886, 3.38129691002454990953104965376, 3.56202432856041743610982345722, 3.82582147463208900487057109343, 4.06470651024197752270995687351, 4.13077921822599670970705716554, 4.33017784552889387201559369052, 4.43686246709029978677315629674, 4.53228677761958922868746905640, 4.61897462494894681333250983222, 4.94900608211865650205103718124

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

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