Properties

Label 2-6240-1.1-c1-0-74
Degree $2$
Conductor $6240$
Sign $-1$
Analytic cond. $49.8266$
Root an. cond. $7.05879$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 3.37·7-s + 9-s + 0.627·11-s − 13-s + 15-s − 5.37·17-s + 6.74·19-s − 3.37·21-s − 7.37·23-s + 25-s − 27-s + 8.74·29-s − 0.627·33-s − 3.37·35-s − 9.37·37-s + 39-s − 8.11·41-s − 10.7·43-s − 45-s − 4·47-s + 4.37·49-s + 5.37·51-s − 1.37·53-s − 0.627·55-s − 6.74·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.27·7-s + 0.333·9-s + 0.189·11-s − 0.277·13-s + 0.258·15-s − 1.30·17-s + 1.54·19-s − 0.735·21-s − 1.53·23-s + 0.200·25-s − 0.192·27-s + 1.62·29-s − 0.109·33-s − 0.570·35-s − 1.54·37-s + 0.160·39-s − 1.26·41-s − 1.63·43-s − 0.149·45-s − 0.583·47-s + 0.624·49-s + 0.752·51-s − 0.188·53-s − 0.0846·55-s − 0.893·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6240\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(49.8266\)
Root analytic conductor: \(7.05879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6240,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 + T \)
good7 \( 1 - 3.37T + 7T^{2} \)
11 \( 1 - 0.627T + 11T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
19 \( 1 - 6.74T + 19T^{2} \)
23 \( 1 + 7.37T + 23T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 9.37T + 37T^{2} \)
41 \( 1 + 8.11T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 2.62T + 61T^{2} \)
67 \( 1 - 9.48T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + 9.37T + 89T^{2} \)
97 \( 1 - 5.37T + 97T^{2} \)
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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

−7.75119508293443106441404280917, −6.91466643496912990542109618319, −6.43010803814193596654875110228, −5.25242386947782923041306580521, −4.96680576556264651035993779605, −4.19451499187938018385984291632, −3.32494160183626309016854614940, −2.11043362052250107576283366476, −1.31679805159273355004191117877, 0, 1.31679805159273355004191117877, 2.11043362052250107576283366476, 3.32494160183626309016854614940, 4.19451499187938018385984291632, 4.96680576556264651035993779605, 5.25242386947782923041306580521, 6.43010803814193596654875110228, 6.91466643496912990542109618319, 7.75119508293443106441404280917

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