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Properties

Label 2-310464-1.1-c1-0-115
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 − 4·13-s − 6·17-s − 2·19-s − 25-s − 6·29-s + 2·31-s − 2·37-s + 2·41-s + 12·43-s + 6·47-s + 10·53-s + 2·55-s + 8·59-s − 4·61-s − 8·65-s + 8·71-s + 6·73-s + 12·79-s + 6·83-s − 12·85-s + 12·89-s − 4·95-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.301·11-s − 1.10·13-s − 1.45·17-s − 0.458·19-s − 1/5·25-s − 1.11·29-s + 0.359·31-s − 0.328·37-s + 0.312·41-s + 1.82·43-s + 0.875·47-s + 1.37·53-s + 0.269·55-s + 1.04·59-s − 0.512·61-s − 0.992·65-s + 0.949·71-s + 0.702·73-s + 1.35·79-s + 0.658·83-s − 1.30·85-s + 1.27·89-s − 0.410·95-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(2.224471574\)
\(L(\frac12)\) \(\approx\) \(2.224471574\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−12.70996570983258, −12.19114650762385, −11.84244325355010, −11.19412430554502, −10.78048755551709, −10.44418451874188, −9.794039809886658, −9.472487664137046, −9.007951470389279, −8.746103412101074, −7.951030458199076, −7.529452064034498, −6.988961030747836, −6.624778015217872, −5.949020558874485, −5.762280249185876, −4.958907812956725, −4.722232280504950, −3.898456343509591, −3.718054512823029, −2.598229490197665, −2.236889868147050, −2.088582723589397, −1.096488441689577, −0.4053259126067686, 0.4053259126067686, 1.096488441689577, 2.088582723589397, 2.236889868147050, 2.598229490197665, 3.718054512823029, 3.898456343509591, 4.722232280504950, 4.958907812956725, 5.762280249185876, 5.949020558874485, 6.624778015217872, 6.988961030747836, 7.529452064034498, 7.951030458199076, 8.746103412101074, 9.007951470389279, 9.472487664137046, 9.794039809886658, 10.44418451874188, 10.78048755551709, 11.19412430554502, 11.84244325355010, 12.19114650762385, 12.70996570983258

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