Properties

Label 2-79560-1.1-c1-0-6
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 − 4·11-s − 13-s − 17-s + 8·19-s + 25-s + 6·29-s − 8·31-s + 2·37-s + 10·41-s + 4·43-s − 7·49-s + 6·53-s − 4·55-s − 6·61-s − 65-s + 4·67-s + 8·71-s − 14·73-s + 4·79-s + 4·83-s − 85-s + 14·89-s + 8·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s − 0.277·13-s − 0.242·17-s + 1.83·19-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.328·37-s + 1.56·41-s + 0.609·43-s − 49-s + 0.824·53-s − 0.539·55-s − 0.768·61-s − 0.124·65-s + 0.488·67-s + 0.949·71-s − 1.63·73-s + 0.450·79-s + 0.439·83-s − 0.108·85-s + 1.48·89-s + 0.820·95-s + 0.203·97-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

\(L(1)\) \(\approx\) \(2.384716132\)
\(L(\frac12)\) \(\approx\) \(2.384716132\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.95230751207737, −13.49315380133422, −13.07701413364390, −12.52114486744633, −12.13025378809754, −11.43752095841343, −10.98411229819020, −10.50365139536634, −9.973816539572452, −9.470163160409858, −9.107016711383828, −8.402074128862591, −7.734274421880387, −7.482814033824121, −6.913025953869330, −6.103220246295859, −5.706423638141940, −5.093930980175111, −4.766589698582666, −3.917326510429364, −3.196542113174415, −2.684011009793633, −2.136816759858057, −1.250440889432108, −0.5274584308394039, 0.5274584308394039, 1.250440889432108, 2.136816759858057, 2.684011009793633, 3.196542113174415, 3.917326510429364, 4.766589698582666, 5.093930980175111, 5.706423638141940, 6.103220246295859, 6.913025953869330, 7.482814033824121, 7.734274421880387, 8.402074128862591, 9.107016711383828, 9.470163160409858, 9.973816539572452, 10.50365139536634, 10.98411229819020, 11.43752095841343, 12.13025378809754, 12.52114486744633, 13.07701413364390, 13.49315380133422, 13.95230751207737

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