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Properties

Label 2-65520-1.1-c1-0-105
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 + 4·11-s + 13-s − 2·17-s + 4·19-s + 25-s + 2·29-s − 8·31-s − 35-s + 6·37-s + 6·41-s − 4·43-s + 49-s − 6·53-s − 4·55-s − 12·59-s − 2·61-s − 65-s − 4·67-s + 10·73-s + 4·77-s + 4·83-s + 2·85-s − 10·89-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 + 0.986·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.824·53-s − 0.539·55-s − 1.56·59-s − 0.256·61-s − 0.124·65-s − 0.488·67-s + 1.17·73-s + 0.455·77-s + 0.439·83-s + 0.216·85-s − 1.05·89-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: \(1\)
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 \( 1 - 4 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.51038432315075, −13.96716572100123, −13.59566260530271, −12.87823710560750, −12.40633460948002, −11.91960829831980, −11.40749520922331, −11.02301549900190, −10.61837742798972, −9.669729148331523, −9.380819394904994, −8.904909434596262, −8.279516216453420, −7.716544804925939, −7.289194202804941, −6.625569667421659, −6.162726604121430, −5.523876514492657, −4.821526384513357, −4.316230686444708, −3.732672276723146, −3.204689963671471, −2.406246961159054, −1.540062460504457, −1.058801592774500, 0, 1.058801592774500, 1.540062460504457, 2.406246961159054, 3.204689963671471, 3.732672276723146, 4.316230686444708, 4.821526384513357, 5.523876514492657, 6.162726604121430, 6.625569667421659, 7.289194202804941, 7.716544804925939, 8.279516216453420, 8.904909434596262, 9.380819394904994, 9.669729148331523, 10.61837742798972, 11.02301549900190, 11.40749520922331, 11.91960829831980, 12.40633460948002, 12.87823710560750, 13.59566260530271, 13.96716572100123, 14.51038432315075

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