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Properties

Label 2-65520-1.1-c1-0-110
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 13-s + 2·17-s + 4·19-s + 25-s + 2·29-s + 35-s + 2·37-s − 6·41-s + 4·43-s − 8·47-s + 49-s − 6·53-s − 4·59-s − 10·61-s − 65-s − 12·67-s + 4·71-s − 10·73-s + 12·83-s + 2·85-s + 18·89-s − 91-s + 4·95-s − 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.277·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.169·35-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.520·59-s − 1.28·61-s − 0.124·65-s − 1.46·67-s + 0.474·71-s − 1.17·73-s + 1.31·83-s + 0.216·85-s + 1.90·89-s − 0.104·91-s + 0.410·95-s − 0.203·97-s + 0.0995·101-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: \(1\)
Selberg data: \((2,\ 65520,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.37120734473784, −14.07154733829302, −13.37750137014345, −13.15330190504126, −12.32443254876095, −11.97342802641778, −11.55862796217585, −10.80847934193878, −10.46708658492357, −9.846616572807287, −9.401698933582616, −8.928242557170811, −8.255335998607794, −7.664917318908083, −7.386697169469936, −6.526173372917169, −6.130669993902785, −5.479991058135149, −4.883303102059870, −4.550045906405627, −3.564307332519977, −3.112567126448597, −2.407227100690200, −1.614963108286217, −1.089572885216528, 0, 1.089572885216528, 1.614963108286217, 2.407227100690200, 3.112567126448597, 3.564307332519977, 4.550045906405627, 4.883303102059870, 5.479991058135149, 6.130669993902785, 6.526173372917169, 7.386697169469936, 7.664917318908083, 8.255335998607794, 8.928242557170811, 9.401698933582616, 9.846616572807287, 10.46708658492357, 10.80847934193878, 11.55862796217585, 11.97342802641778, 12.32443254876095, 13.15330190504126, 13.37750137014345, 14.07154733829302, 14.37120734473784

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