Properties

Label 2-310464-1.1-c1-0-22
Degree $2$
Conductor $310464$
Sign $1$
Analytic cond. $2479.06$
Root an. cond. $49.7902$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 11-s − 6·13-s − 2·17-s + 8·19-s − 4·23-s − 25-s + 2·29-s − 8·31-s − 6·37-s − 2·41-s + 8·43-s + 4·47-s + 2·53-s + 2·55-s − 12·59-s + 10·61-s + 12·65-s − 12·67-s − 12·71-s − 10·73-s + 8·79-s + 12·83-s + 4·85-s + 10·89-s − 16·95-s + 14·97-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.301·11-s − 1.66·13-s − 0.485·17-s + 1.83·19-s − 0.834·23-s − 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.986·37-s − 0.312·41-s + 1.21·43-s + 0.583·47-s + 0.274·53-s + 0.269·55-s − 1.56·59-s + 1.28·61-s + 1.48·65-s − 1.46·67-s − 1.42·71-s − 1.17·73-s + 0.900·79-s + 1.31·83-s + 0.433·85-s + 1.05·89-s − 1.64·95-s + 1.42·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310464\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(2479.06\)
Root analytic conductor: \(49.7902\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 310464,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5343149633\)
\(L(\frac12)\) \(\approx\) \(0.5343149633\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.37916782055829, −12.14982838339049, −11.88869146419793, −11.41435210952895, −10.83600655551141, −10.32436443662922, −10.00808198334878, −9.322977447589874, −9.118411127994383, −8.505242277884166, −7.739631265085301, −7.566238087948747, −7.306487960317827, −6.776106925545397, −5.862769705555487, −5.673765997267915, −4.887658923466727, −4.678891292929921, −4.031621314791035, −3.393505185503608, −3.094779356659490, −2.275266654614972, −1.914486965118006, −0.9741743811477432, −0.2178492054879602, 0.2178492054879602, 0.9741743811477432, 1.914486965118006, 2.275266654614972, 3.094779356659490, 3.393505185503608, 4.031621314791035, 4.678891292929921, 4.887658923466727, 5.673765997267915, 5.862769705555487, 6.776106925545397, 7.306487960317827, 7.566238087948747, 7.739631265085301, 8.505242277884166, 9.118411127994383, 9.322977447589874, 10.00808198334878, 10.32436443662922, 10.83600655551141, 11.41435210952895, 11.88869146419793, 12.14982838339049, 12.37916782055829

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