Properties

Label 2-115-1.1-c5-0-24
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  + 8.26·2-s + 12.5·3-s + 36.2·4-s + 25·5-s + 104.·6-s + 63.2·7-s + 35.1·8-s − 84.3·9-s + 206.·10-s + 468.·11-s + 456.·12-s + 1.12e3·13-s + 522.·14-s + 314.·15-s − 869.·16-s − 601.·17-s − 696.·18-s − 608.·19-s + 906.·20-s + 796.·21-s + 3.87e3·22-s − 529·23-s + 442.·24-s + 625·25-s + 9.27e3·26-s − 4.12e3·27-s + 2.29e3·28-s + ⋯
L(s)  = 1  + 1.46·2-s + 0.807·3-s + 1.13·4-s + 0.447·5-s + 1.17·6-s + 0.487·7-s + 0.193·8-s − 0.347·9-s + 0.653·10-s + 1.16·11-s + 0.915·12-s + 1.84·13-s + 0.712·14-s + 0.361·15-s − 0.849·16-s − 0.504·17-s − 0.507·18-s − 0.386·19-s + 0.506·20-s + 0.394·21-s + 1.70·22-s − 0.208·23-s + 0.156·24-s + 0.200·25-s + 2.69·26-s − 1.08·27-s + 0.552·28-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

\(L(3)\) \(\approx\) \(5.798454209\)
\(L(\frac12)\) \(\approx\) \(5.798454209\)
\(L(\frac{7}{2})\) 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 \( 1 - 8.26T + 32T^{2} \)
3 \( 1 - 12.5T + 243T^{2} \)
7 \( 1 - 63.2T + 1.68e4T^{2} \)
11 \( 1 - 468.T + 1.61e5T^{2} \)
13 \( 1 - 1.12e3T + 3.71e5T^{2} \)
17 \( 1 + 601.T + 1.41e6T^{2} \)
19 \( 1 + 608.T + 2.47e6T^{2} \)
29 \( 1 - 1.89e3T + 2.05e7T^{2} \)
31 \( 1 + 4.45e3T + 2.86e7T^{2} \)
37 \( 1 - 134.T + 6.93e7T^{2} \)
41 \( 1 + 8.54e3T + 1.15e8T^{2} \)
43 \( 1 - 1.46e3T + 1.47e8T^{2} \)
47 \( 1 - 4.43e3T + 2.29e8T^{2} \)
53 \( 1 + 1.33e4T + 4.18e8T^{2} \)
59 \( 1 - 4.84e4T + 7.14e8T^{2} \)
61 \( 1 + 3.76e4T + 8.44e8T^{2} \)
67 \( 1 + 1.21e4T + 1.35e9T^{2} \)
71 \( 1 + 6.78e4T + 1.80e9T^{2} \)
73 \( 1 - 6.90e4T + 2.07e9T^{2} \)
79 \( 1 + 7.56e4T + 3.07e9T^{2} \)
83 \( 1 - 5.78e4T + 3.93e9T^{2} \)
89 \( 1 + 7.00e4T + 5.58e9T^{2} \)
97 \( 1 - 1.00e4T + 8.58e9T^{2} \)
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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

−13.06209095553742599980563855271, −11.76111481515960880635121470523, −10.96929301776223921314602282501, −9.160183944484929070952685294093, −8.436938589101688658367478028547, −6.61296004860623047775555827151, −5.73158190451318165764207592731, −4.22560003337489821682573459456, −3.28441480810567847629992578349, −1.78226617409113099548997557121, 1.78226617409113099548997557121, 3.28441480810567847629992578349, 4.22560003337489821682573459456, 5.73158190451318165764207592731, 6.61296004860623047775555827151, 8.436938589101688658367478028547, 9.160183944484929070952685294093, 10.96929301776223921314602282501, 11.76111481515960880635121470523, 13.06209095553742599980563855271

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