Properties

Label 4-39e4-1.1-c3e2-0-1
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $8053.60$
Root an. cond. $9.47322$
Motivic weight $3$
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  − 16·4-s + 192·16-s − 234·17-s + 36·23-s − 223·25-s + 198·29-s + 164·43-s − 578·49-s + 522·53-s − 1.43e3·61-s − 2.04e3·64-s + 3.74e3·68-s − 880·79-s − 576·92-s + 3.56e3·100-s + 3.15e3·101-s − 1.58e3·103-s − 900·107-s + 3.40e3·113-s − 3.16e3·116-s + 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 2·4-s + 3·16-s − 3.33·17-s + 0.326·23-s − 1.78·25-s + 1.26·29-s + 0.581·43-s − 1.68·49-s + 1.35·53-s − 3.01·61-s − 4·64-s + 6.67·68-s − 1.25·79-s − 0.652·92-s + 3.56·100-s + 3.10·101-s − 1.51·103-s − 0.813·107-s + 2.83·113-s − 2.53·116-s + 0.0285·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(8053.60\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9127312975\)
\(L(\frac12)\) \(\approx\) \(0.9127312975\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
5$C_2^2$ \( 1 + 223 T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 578 T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 38 T^{2} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 117 T + p^{3} T^{2} )^{2} \)
19$C_2^2$ \( 1 + 13130 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 18 T + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 99 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 21950 T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 88631 T^{2} + p^{6} T^{4} \)
41$C_2^2$ \( 1 + 136519 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 - 82 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 202354 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 261 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 213050 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 + 719 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 107018 T^{2} + p^{6} T^{4} \)
71$C_2^2$ \( 1 + 497122 T^{2} + p^{6} T^{4} \)
73$C_2^2$ \( 1 + 309959 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 440 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 284726 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 - 892190 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 + 486674 T^{2} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.196855160486392250069417837272, −8.875680978922095134004280533097, −8.579848325042716077864746150102, −8.393941427481593426623831459224, −7.64271061413475533377782181775, −7.59716480120437699781270520381, −6.68409782970116432114698550067, −6.56612766689132293110780450780, −5.80275354925344234412552371249, −5.73233614353305891516211688169, −4.93660634141657034013746439986, −4.48331616707039185470501519802, −4.38584372684602729103394242428, −4.14140804317027894772438440768, −3.23672680049020394011929417960, −3.01002074647692909532497695576, −2.00637084948877784728035379564, −1.74125818581573585537240930160, −0.61478563863749402545826123056, −0.36853556180479021846945288883, 0.36853556180479021846945288883, 0.61478563863749402545826123056, 1.74125818581573585537240930160, 2.00637084948877784728035379564, 3.01002074647692909532497695576, 3.23672680049020394011929417960, 4.14140804317027894772438440768, 4.38584372684602729103394242428, 4.48331616707039185470501519802, 4.93660634141657034013746439986, 5.73233614353305891516211688169, 5.80275354925344234412552371249, 6.56612766689132293110780450780, 6.68409782970116432114698550067, 7.59716480120437699781270520381, 7.64271061413475533377782181775, 8.393941427481593426623831459224, 8.579848325042716077864746150102, 8.875680978922095134004280533097, 9.196855160486392250069417837272

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