Properties

Label 2-115-1.1-c3-0-3
Degree $2$
Conductor $115$
Sign $1$
Analytic cond. $6.78521$
Root an. cond. $2.60484$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.404·2-s − 7.11·3-s − 7.83·4-s − 5·5-s − 2.88·6-s + 13.7·7-s − 6.41·8-s + 23.5·9-s − 2.02·10-s + 24.2·11-s + 55.7·12-s + 3.05·13-s + 5.58·14-s + 35.5·15-s + 60.0·16-s + 63.1·17-s + 9.55·18-s − 2.07·19-s + 39.1·20-s − 98.0·21-s + 9.81·22-s − 23·23-s + 45.6·24-s + 25·25-s + 1.23·26-s + 24.1·27-s − 108.·28-s + ⋯
L(s)  = 1  + 0.143·2-s − 1.36·3-s − 0.979·4-s − 0.447·5-s − 0.195·6-s + 0.744·7-s − 0.283·8-s + 0.874·9-s − 0.0640·10-s + 0.664·11-s + 1.34·12-s + 0.0650·13-s + 0.106·14-s + 0.612·15-s + 0.938·16-s + 0.900·17-s + 0.125·18-s − 0.0250·19-s + 0.438·20-s − 1.01·21-s + 0.0950·22-s − 0.208·23-s + 0.387·24-s + 0.200·25-s + 0.00931·26-s + 0.172·27-s − 0.729·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $1$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7968129856\)
\(L(\frac12)\) \(\approx\) \(0.7968129856\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
23 \( 1 + 23T \)
good2 \( 1 - 0.404T + 8T^{2} \)
3 \( 1 + 7.11T + 27T^{2} \)
7 \( 1 - 13.7T + 343T^{2} \)
11 \( 1 - 24.2T + 1.33e3T^{2} \)
13 \( 1 - 3.05T + 2.19e3T^{2} \)
17 \( 1 - 63.1T + 4.91e3T^{2} \)
19 \( 1 + 2.07T + 6.85e3T^{2} \)
29 \( 1 + 8.16T + 2.43e4T^{2} \)
31 \( 1 + 156.T + 2.97e4T^{2} \)
37 \( 1 - 302.T + 5.06e4T^{2} \)
41 \( 1 + 42.7T + 6.89e4T^{2} \)
43 \( 1 - 215.T + 7.95e4T^{2} \)
47 \( 1 - 247.T + 1.03e5T^{2} \)
53 \( 1 - 600.T + 1.48e5T^{2} \)
59 \( 1 - 92.2T + 2.05e5T^{2} \)
61 \( 1 - 532.T + 2.26e5T^{2} \)
67 \( 1 - 30.3T + 3.00e5T^{2} \)
71 \( 1 + 736.T + 3.57e5T^{2} \)
73 \( 1 - 349.T + 3.89e5T^{2} \)
79 \( 1 - 301.T + 4.93e5T^{2} \)
83 \( 1 - 139.T + 5.71e5T^{2} \)
89 \( 1 - 859.T + 7.04e5T^{2} \)
97 \( 1 + 927.T + 9.12e5T^{2} \)
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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

−12.83666972575607205142439070993, −11.97285188975229984098562489850, −11.23326760976800018727989787211, −10.09465116430192872428776210349, −8.803569236473882793099149441612, −7.55812410833966732413803671307, −5.99402405661476908645184839817, −5.04726139812454266012391483942, −3.95782787608750176330015477855, −0.843850271307433494817457492963, 0.843850271307433494817457492963, 3.95782787608750176330015477855, 5.04726139812454266012391483942, 5.99402405661476908645184839817, 7.55812410833966732413803671307, 8.803569236473882793099149441612, 10.09465116430192872428776210349, 11.23326760976800018727989787211, 11.97285188975229984098562489850, 12.83666972575607205142439070993

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