Properties

Label 4-63e2-1.1-c6e2-0-3
Degree $4$
Conductor $3969$
Sign $1$
Analytic cond. $210.058$
Root an. cond. $3.80702$
Motivic weight $6$
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  + 47·4-s + 686·7-s − 1.88e3·16-s + 3.12e4·25-s + 3.22e4·28-s − 2.02e5·37-s + 2.53e5·43-s + 3.52e5·49-s − 2.81e5·64-s − 1.07e5·67-s + 1.85e6·79-s + 1.46e6·100-s + 5.17e6·109-s − 1.29e6·112-s − 3.06e5·121-s + 127-s + 131-s + 137-s + 139-s − 9.51e6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.65e6·169-s + 1.19e7·172-s + ⋯
L(s)  = 1  + 0.734·4-s + 2·7-s − 0.460·16-s + 2·25-s + 1.46·28-s − 3.99·37-s + 3.18·43-s + 3·49-s − 1.07·64-s − 0.358·67-s + 3.76·79-s + 1.46·100-s + 3.99·109-s − 0.921·112-s − 0.172·121-s − 2.93·148-s + 2·169-s + 2.33·172-s + 4·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(210.058\)
Root analytic conductor: \(3.80702\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3969,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(4.109686704\)
\(L(\frac12)\) \(\approx\) \(4.109686704\)
\(L(4)\) 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 \( 1 \)
7$C_1$ \( ( 1 - p^{3} T )^{2} \)
good2$C_2^2$ \( 1 - 47 T^{2} + p^{12} T^{4} \)
5$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
11$C_2^2$ \( 1 + 306322 T^{2} + p^{12} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
23$C_2^2$ \( 1 + 220762978 T^{2} + p^{12} T^{4} \)
29$C_2^2$ \( 1 - 739273358 T^{2} + p^{12} T^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
37$C_2$ \( ( 1 + 101194 T + p^{6} T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
43$C_2$ \( ( 1 - 126614 T + p^{6} T^{2} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
53$C_2^2$ \( 1 - 41794002542 T^{2} + p^{12} T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
67$C_2$ \( ( 1 + 53926 T + p^{6} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 197404987358 T^{2} + p^{12} T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
79$C_2$ \( ( 1 - 929378 T + p^{6} T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \)
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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

−13.99603602666454354258778960168, −13.82243808280328853624588583225, −12.65352192991361697573664450396, −12.23927513550964172339226775758, −11.77698985870437320286221923644, −10.90379413150738443854923675921, −10.89474288179193132384352987289, −10.36073701704036897170190476544, −8.991542906948289853724439826466, −8.864622976596343470673342150727, −8.071467396659716318957731530126, −7.29062663356012248787780563648, −7.00876962171962075367907818100, −5.99478072518067221030114826916, −5.06292275426921757581100861538, −4.74890175930997856568266779860, −3.63248880422119805175091805606, −2.44036024482349478053666864062, −1.79149478733878555733347850535, −0.865858192482725179500171759132, 0.865858192482725179500171759132, 1.79149478733878555733347850535, 2.44036024482349478053666864062, 3.63248880422119805175091805606, 4.74890175930997856568266779860, 5.06292275426921757581100861538, 5.99478072518067221030114826916, 7.00876962171962075367907818100, 7.29062663356012248787780563648, 8.071467396659716318957731530126, 8.864622976596343470673342150727, 8.991542906948289853724439826466, 10.36073701704036897170190476544, 10.89474288179193132384352987289, 10.90379413150738443854923675921, 11.77698985870437320286221923644, 12.23927513550964172339226775758, 12.65352192991361697573664450396, 13.82243808280328853624588583225, 13.99603602666454354258778960168

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