Properties

Label 2-315-1.1-c1-0-5
Degree $2$
Conductor $315$
Sign $-1$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  − 2-s − 4-s − 5-s + 7-s + 3·8-s + 10-s − 6·13-s − 14-s − 16-s − 2·17-s − 8·19-s + 20-s − 8·23-s + 25-s + 6·26-s − 28-s + 2·29-s + 4·31-s − 5·32-s + 2·34-s − 35-s − 2·37-s + 8·38-s − 3·40-s + 6·41-s + 4·43-s + 8·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.316·10-s − 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s − 1.83·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.371·29-s + 0.718·31-s − 0.883·32-s + 0.342·34-s − 0.169·35-s − 0.328·37-s + 1.29·38-s − 0.474·40-s + 0.937·41-s + 0.609·43-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{315} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−10.96971784880108147181434505980, −10.15928312968820765258481052050, −9.326498061922569555987691090592, −8.261177977090796325600765542566, −7.73831918709431562684545235915, −6.48451110526696282246961711325, −4.87484917945516723546545158294, −4.17084275561163201389021211529, −2.17599002568151719086793905114, 0, 2.17599002568151719086793905114, 4.17084275561163201389021211529, 4.87484917945516723546545158294, 6.48451110526696282246961711325, 7.73831918709431562684545235915, 8.261177977090796325600765542566, 9.326498061922569555987691090592, 10.15928312968820765258481052050, 10.96971784880108147181434505980

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