Properties

Label 2-130-1.1-c1-0-0
Degree $2$
Conductor $130$
Sign $1$
Analytic cond. $1.03805$
Root an. cond. $1.01884$
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  + 2-s + 4-s + 5-s + 8-s − 3·9-s + 10-s + 13-s + 16-s + 2·17-s − 3·18-s − 8·19-s + 20-s − 4·23-s + 25-s + 26-s − 2·29-s − 4·31-s + 32-s + 2·34-s − 3·36-s + 6·37-s − 8·38-s + 40-s + 10·41-s − 3·45-s − 4·46-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 9-s + 0.316·10-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 1.83·19-s + 0.223·20-s − 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 0.986·37-s − 1.29·38-s + 0.158·40-s + 1.56·41-s − 0.447·45-s − 0.589·46-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.564713754\)
\(L(\frac12)\) \(\approx\) \(1.564713754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.31482683467549778702408058264, −12.46224067291033080181015165159, −11.33556569842126116576011253797, −10.46707546118065350803717010462, −9.084106232831126981543122302400, −7.923578663951581954529383102497, −6.37597525425947166446541877091, −5.56905987734202508880056372110, −4.02639003808572429368239555281, −2.40292181514948168945129765801, 2.40292181514948168945129765801, 4.02639003808572429368239555281, 5.56905987734202508880056372110, 6.37597525425947166446541877091, 7.923578663951581954529383102497, 9.084106232831126981543122302400, 10.46707546118065350803717010462, 11.33556569842126116576011253797, 12.46224067291033080181015165159, 13.31482683467549778702408058264

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