Properties

Label 2-315-1.1-c3-0-27
Degree $2$
Conductor $315$
Sign $-1$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.37·2-s + 3.42·4-s − 5·5-s + 7·7-s − 15.4·8-s − 16.8·10-s − 45.0·11-s − 35.4·13-s + 23.6·14-s − 79.6·16-s − 29.4·17-s + 3.18·19-s − 17.1·20-s − 152.·22-s + 23.6·23-s + 25·25-s − 119.·26-s + 23.9·28-s − 9.22·29-s − 80.2·31-s − 145.·32-s − 99.5·34-s − 35·35-s − 61.1·37-s + 10.7·38-s + 77.3·40-s − 282.·41-s + ⋯
L(s)  = 1  + 1.19·2-s + 0.427·4-s − 0.447·5-s + 0.377·7-s − 0.683·8-s − 0.534·10-s − 1.23·11-s − 0.757·13-s + 0.451·14-s − 1.24·16-s − 0.420·17-s + 0.0384·19-s − 0.191·20-s − 1.47·22-s + 0.214·23-s + 0.200·25-s − 0.904·26-s + 0.161·28-s − 0.0590·29-s − 0.464·31-s − 0.803·32-s − 0.501·34-s − 0.169·35-s − 0.271·37-s + 0.0460·38-s + 0.305·40-s − 1.07·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
7 \( 1 - 7T \)
good2 \( 1 - 3.37T + 8T^{2} \)
11 \( 1 + 45.0T + 1.33e3T^{2} \)
13 \( 1 + 35.4T + 2.19e3T^{2} \)
17 \( 1 + 29.4T + 4.91e3T^{2} \)
19 \( 1 - 3.18T + 6.85e3T^{2} \)
23 \( 1 - 23.6T + 1.21e4T^{2} \)
29 \( 1 + 9.22T + 2.43e4T^{2} \)
31 \( 1 + 80.2T + 2.97e4T^{2} \)
37 \( 1 + 61.1T + 5.06e4T^{2} \)
41 \( 1 + 282.T + 6.89e4T^{2} \)
43 \( 1 + 58.8T + 7.95e4T^{2} \)
47 \( 1 + 371.T + 1.03e5T^{2} \)
53 \( 1 - 256.T + 1.48e5T^{2} \)
59 \( 1 + 571.T + 2.05e5T^{2} \)
61 \( 1 - 835.T + 2.26e5T^{2} \)
67 \( 1 - 933.T + 3.00e5T^{2} \)
71 \( 1 + 378.T + 3.57e5T^{2} \)
73 \( 1 + 494.T + 3.89e5T^{2} \)
79 \( 1 - 1.07e3T + 4.93e5T^{2} \)
83 \( 1 - 722.T + 5.71e5T^{2} \)
89 \( 1 + 89.5T + 7.04e5T^{2} \)
97 \( 1 + 101.T + 9.12e5T^{2} \)
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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

−11.06806138881812740786814399272, −9.980385867617794772318341819833, −8.753570404130965801157535877409, −7.76003949676780028202461263062, −6.68539676999222449378790441002, −5.33326573362356420630163474537, −4.78819254600315534514037656028, −3.56151943909063911641643970271, −2.39149528941306855104739166130, 0, 2.39149528941306855104739166130, 3.56151943909063911641643970271, 4.78819254600315534514037656028, 5.33326573362356420630163474537, 6.68539676999222449378790441002, 7.76003949676780028202461263062, 8.753570404130965801157535877409, 9.980385867617794772318341819833, 11.06806138881812740786814399272

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