Properties

Label 2-115-1.1-c5-0-9
Degree $2$
Conductor $115$
Sign $1$
Analytic cond. $18.4441$
Root an. cond. $4.29466$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.26·2-s − 11.8·3-s − 4.24·4-s − 25·5-s − 62.3·6-s + 72.8·7-s − 190.·8-s − 102.·9-s − 131.·10-s + 732.·11-s + 50.2·12-s + 676.·13-s + 383.·14-s + 296.·15-s − 870.·16-s + 381.·17-s − 541.·18-s + 853.·19-s + 106.·20-s − 862.·21-s + 3.86e3·22-s + 529·23-s + 2.26e3·24-s + 625·25-s + 3.56e3·26-s + 4.09e3·27-s − 309.·28-s + ⋯
L(s)  = 1  + 0.931·2-s − 0.759·3-s − 0.132·4-s − 0.447·5-s − 0.707·6-s + 0.562·7-s − 1.05·8-s − 0.423·9-s − 0.416·10-s + 1.82·11-s + 0.100·12-s + 1.10·13-s + 0.523·14-s + 0.339·15-s − 0.849·16-s + 0.320·17-s − 0.393·18-s + 0.542·19-s + 0.0593·20-s − 0.426·21-s + 1.70·22-s + 0.208·23-s + 0.801·24-s + 0.200·25-s + 1.03·26-s + 1.08·27-s − 0.0745·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $1$
Analytic conductor: \(18.4441\)
Root analytic conductor: \(4.29466\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.034062670\)
\(L(\frac12)\) \(\approx\) \(2.034062670\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25T \)
23 \( 1 - 529T \)
good2 \( 1 - 5.26T + 32T^{2} \)
3 \( 1 + 11.8T + 243T^{2} \)
7 \( 1 - 72.8T + 1.68e4T^{2} \)
11 \( 1 - 732.T + 1.61e5T^{2} \)
13 \( 1 - 676.T + 3.71e5T^{2} \)
17 \( 1 - 381.T + 1.41e6T^{2} \)
19 \( 1 - 853.T + 2.47e6T^{2} \)
29 \( 1 + 3.16e3T + 2.05e7T^{2} \)
31 \( 1 - 6.80e3T + 2.86e7T^{2} \)
37 \( 1 + 2.63e3T + 6.93e7T^{2} \)
41 \( 1 - 1.52e4T + 1.15e8T^{2} \)
43 \( 1 + 6.96e3T + 1.47e8T^{2} \)
47 \( 1 - 8.07e3T + 2.29e8T^{2} \)
53 \( 1 + 1.32e3T + 4.18e8T^{2} \)
59 \( 1 + 2.01e3T + 7.14e8T^{2} \)
61 \( 1 + 1.96e4T + 8.44e8T^{2} \)
67 \( 1 - 2.37e4T + 1.35e9T^{2} \)
71 \( 1 - 4.44e4T + 1.80e9T^{2} \)
73 \( 1 - 9.20e3T + 2.07e9T^{2} \)
79 \( 1 - 2.76e4T + 3.07e9T^{2} \)
83 \( 1 - 3.25e4T + 3.93e9T^{2} \)
89 \( 1 + 1.17e5T + 5.58e9T^{2} \)
97 \( 1 + 2.83e4T + 8.58e9T^{2} \)
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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

−12.45387879019283832476018628264, −11.68543693155619876969845637269, −11.13094798772389060334564866315, −9.342892799258983729581692605775, −8.351960033095260838349684934136, −6.58489048974875393824396593684, −5.69925717033744538983798306240, −4.48114204936046289456726908743, −3.45006378026906794262316038185, −0.957774225339019551451002904291, 0.957774225339019551451002904291, 3.45006378026906794262316038185, 4.48114204936046289456726908743, 5.69925717033744538983798306240, 6.58489048974875393824396593684, 8.351960033095260838349684934136, 9.342892799258983729581692605775, 11.13094798772389060334564866315, 11.68543693155619876969845637269, 12.45387879019283832476018628264

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