Properties

Label 12-840e6-1.1-c1e6-0-1
Degree $12$
Conductor $3.513\times 10^{17}$
Sign $1$
Analytic cond. $91062.2$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6}\)
Sign: $1$
Analytic conductor: \(91062.2\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{18} \cdot 3^{6} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.03394988590\)
\(L(\frac12)\) \(\approx\) \(0.03394988590\)
\(L(\frac{3}{2})\) 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

−5.33952927863097609466118634686, −5.30320136825486485088160033048, −5.24909029607056907244009885272, −4.96084861583235354830568164522, −4.81998918884928241872155186892, −4.67591968158364271176945699112, −4.36260373509215571501202023228, −4.34274330840527821330786408653, −3.89318873789888755691836913592, −3.73835875176629263664898565403, −3.72663849551707956447343889321, −3.54130857908522219542163562190, −3.49423281154883589812273604943, −3.03815599890933320957317920990, −2.94387350978937265984482804805, −2.77949748108983990780820873838, −2.69414588236829131715019694196, −2.28922198302517342123839703834, −2.11412961738966274410127478952, −1.94983708489767539875044730208, −1.41841607886830970650161220133, −1.36151067497188397762866344712, −1.22736369760018781660192023016, −0.39833095299998878180378607508, −0.05465619105698653261970049521, 0.05465619105698653261970049521, 0.39833095299998878180378607508, 1.22736369760018781660192023016, 1.36151067497188397762866344712, 1.41841607886830970650161220133, 1.94983708489767539875044730208, 2.11412961738966274410127478952, 2.28922198302517342123839703834, 2.69414588236829131715019694196, 2.77949748108983990780820873838, 2.94387350978937265984482804805, 3.03815599890933320957317920990, 3.49423281154883589812273604943, 3.54130857908522219542163562190, 3.72663849551707956447343889321, 3.73835875176629263664898565403, 3.89318873789888755691836913592, 4.34274330840527821330786408653, 4.36260373509215571501202023228, 4.67591968158364271176945699112, 4.81998918884928241872155186892, 4.96084861583235354830568164522, 5.24909029607056907244009885272, 5.30320136825486485088160033048, 5.33952927863097609466118634686

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

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