Properties

Label 2-310464-1.1-c1-0-107
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 + 2·13-s − 3·17-s − 7·19-s − 7·23-s − 25-s − 5·29-s + 2·31-s − 3·37-s + 6·41-s + 11·43-s + 7·47-s − 4·53-s + 2·55-s + 11·59-s + 10·61-s + 4·65-s − 4·67-s − 5·71-s + 8·73-s + 8·79-s + 14·83-s − 6·85-s + 2·89-s − 14·95-s − 15·97-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.301·11-s + 0.554·13-s − 0.727·17-s − 1.60·19-s − 1.45·23-s − 1/5·25-s − 0.928·29-s + 0.359·31-s − 0.493·37-s + 0.937·41-s + 1.67·43-s + 1.02·47-s − 0.549·53-s + 0.269·55-s + 1.43·59-s + 1.28·61-s + 0.496·65-s − 0.488·67-s − 0.593·71-s + 0.936·73-s + 0.900·79-s + 1.53·83-s − 0.650·85-s + 0.211·89-s − 1.43·95-s − 1.52·97-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

\(L(1)\) \(\approx\) \(2.222686392\)
\(L(\frac12)\) \(\approx\) \(2.222686392\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−12.70141659234930, −12.24170963351099, −11.82096758424169, −11.07950682582792, −10.86136685004897, −10.45381584518413, −9.851133399654097, −9.452066779967904, −9.020057866911902, −8.599970105363708, −8.034263142182938, −7.665297145116965, −6.878481504476083, −6.492282495368364, −6.127008263884077, −5.617958346717125, −5.281814343008364, −4.274658932285395, −4.127937889938476, −3.700308542519119, −2.700486387716396, −2.180923597992556, −1.987451808861945, −1.177072933636803, −0.3913450652745051, 0.3913450652745051, 1.177072933636803, 1.987451808861945, 2.180923597992556, 2.700486387716396, 3.700308542519119, 4.127937889938476, 4.274658932285395, 5.281814343008364, 5.617958346717125, 6.127008263884077, 6.492282495368364, 6.878481504476083, 7.665297145116965, 8.034263142182938, 8.599970105363708, 9.020057866911902, 9.452066779967904, 9.851133399654097, 10.45381584518413, 10.86136685004897, 11.07950682582792, 11.82096758424169, 12.24170963351099, 12.70141659234930

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