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Properties

Label 2-65520-1.1-c1-0-29
Degree $2$
Conductor $65520$
Sign $1$
Analytic cond. $523.179$
Root an. cond. $22.8731$
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  − 5-s − 7-s + 4·11-s + 13-s − 2·17-s + 4·19-s + 25-s + 2·29-s − 8·31-s + 35-s − 2·37-s + 6·41-s + 4·43-s + 49-s + 2·53-s − 4·55-s + 12·59-s + 14·61-s − 65-s − 4·67-s + 10·73-s − 4·77-s − 12·83-s + 2·85-s + 6·89-s − 91-s − 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.169·35-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.274·53-s − 0.539·55-s + 1.56·59-s + 1.79·61-s − 0.124·65-s − 0.488·67-s + 1.17·73-s − 0.455·77-s − 1.31·83-s + 0.216·85-s + 0.635·89-s − 0.104·91-s − 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65520\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(523.179\)
Root analytic conductor: \(22.8731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 65520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.293710188\)
\(L(\frac12)\) \(\approx\) \(2.293710188\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.18786471527851, −13.82151270823328, −13.14403537801263, −12.68764205226035, −12.20222106726283, −11.66214349719180, −11.22374752577976, −10.84182817857895, −10.05480045948250, −9.580790875240762, −9.094132395924368, −8.659193322200943, −8.058189168172852, −7.379802308969122, −6.872952749955250, −6.574403268643740, −5.656209789100145, −5.417659316717748, −4.442091570944123, −3.957803332154444, −3.551764011279911, −2.811011782787074, −2.054137071297825, −1.221209922100026, −0.5603601564502979, 0.5603601564502979, 1.221209922100026, 2.054137071297825, 2.811011782787074, 3.551764011279911, 3.957803332154444, 4.442091570944123, 5.417659316717748, 5.656209789100145, 6.574403268643740, 6.872952749955250, 7.379802308969122, 8.058189168172852, 8.659193322200943, 9.094132395924368, 9.580790875240762, 10.05480045948250, 10.84182817857895, 11.22374752577976, 11.66214349719180, 12.20222106726283, 12.68764205226035, 13.14403537801263, 13.82151270823328, 14.18786471527851

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