Properties

Label 2-79560-1.1-c1-0-0
Degree $2$
Conductor $79560$
Sign $1$
Analytic cond. $635.289$
Root an. cond. $25.2049$
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 − 2·11-s − 13-s + 17-s − 8·19-s − 6·23-s + 25-s − 4·31-s + 2·37-s − 4·41-s − 4·43-s − 7·49-s + 6·53-s − 2·55-s − 4·59-s + 8·61-s − 65-s + 8·67-s + 4·71-s + 10·79-s − 8·83-s + 85-s − 6·89-s − 8·95-s + 8·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.603·11-s − 0.277·13-s + 0.242·17-s − 1.83·19-s − 1.25·23-s + 1/5·25-s − 0.718·31-s + 0.328·37-s − 0.624·41-s − 0.609·43-s − 49-s + 0.824·53-s − 0.269·55-s − 0.520·59-s + 1.02·61-s − 0.124·65-s + 0.977·67-s + 0.474·71-s + 1.12·79-s − 0.878·83-s + 0.108·85-s − 0.635·89-s − 0.820·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(79560\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(635.289\)
Root analytic conductor: \(25.2049\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{79560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 79560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9578361791\)
\(L(\frac12)\) \(\approx\) \(0.9578361791\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.13984522565794, −13.32150655561149, −13.13249866324483, −12.46479242713400, −12.23240575149508, −11.36442956712504, −11.07009073094496, −10.35659225089561, −10.05843076534120, −9.628084536830404, −8.841231378214705, −8.463889336577581, −7.935744882981305, −7.428427267976677, −6.601308832915235, −6.393788208852618, −5.659855275567428, −5.185467192925619, −4.580773773964677, −3.931834520413895, −3.403592170749322, −2.427099977918653, −2.178694021972665, −1.422462789976538, −0.3037766695015894, 0.3037766695015894, 1.422462789976538, 2.178694021972665, 2.427099977918653, 3.403592170749322, 3.931834520413895, 4.580773773964677, 5.185467192925619, 5.659855275567428, 6.393788208852618, 6.601308832915235, 7.428427267976677, 7.935744882981305, 8.463889336577581, 8.841231378214705, 9.628084536830404, 10.05843076534120, 10.35659225089561, 11.07009073094496, 11.36442956712504, 12.23240575149508, 12.46479242713400, 13.13249866324483, 13.32150655561149, 14.13984522565794

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