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Properties

Label 2-65520-1.1-c1-0-7
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 + 6·19-s − 2·23-s + 25-s − 6·29-s + 8·31-s − 35-s − 6·37-s + 8·41-s − 4·43-s − 8·47-s + 49-s + 4·55-s − 10·59-s − 14·61-s + 65-s + 4·67-s + 6·71-s + 10·73-s − 4·77-s − 16·79-s + 16·89-s − 91-s − 6·95-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 1.20·11-s − 0.277·13-s + 1.37·19-s − 0.417·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.169·35-s − 0.986·37-s + 1.24·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.539·55-s − 1.30·59-s − 1.79·61-s + 0.124·65-s + 0.488·67-s + 0.712·71-s + 1.17·73-s − 0.455·77-s − 1.80·79-s + 1.69·89-s − 0.104·91-s − 0.615·95-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

\(L(1)\) \(\approx\) \(1.279163050\)
\(L(\frac12)\) \(\approx\) \(1.279163050\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−14.23960758528930, −13.61085764900073, −13.34623273233086, −12.66732000477885, −12.06041890757684, −11.85679063754080, −11.10678284506164, −10.75494311905236, −10.20740135238623, −9.552972191275913, −9.260960880266346, −8.338813740678771, −7.976184771420234, −7.622948691303720, −7.069418242177903, −6.371728816756861, −5.680632437420883, −5.166399873418162, −4.733618345256809, −4.070616669641907, −3.253183915376422, −2.886056588444425, −2.064859148498100, −1.329455086955345, −0.3869714629999325, 0.3869714629999325, 1.329455086955345, 2.064859148498100, 2.886056588444425, 3.253183915376422, 4.070616669641907, 4.733618345256809, 5.166399873418162, 5.680632437420883, 6.371728816756861, 7.069418242177903, 7.622948691303720, 7.976184771420234, 8.338813740678771, 9.260960880266346, 9.552972191275913, 10.20740135238623, 10.75494311905236, 11.10678284506164, 11.85679063754080, 12.06041890757684, 12.66732000477885, 13.34623273233086, 13.61085764900073, 14.23960758528930

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