Properties

Label 2-115-1.1-c5-0-27
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  + 9.71·2-s + 18.9·3-s + 62.2·4-s − 25·5-s + 184.·6-s + 84.0·7-s + 294.·8-s + 117.·9-s − 242.·10-s + 219.·11-s + 1.18e3·12-s − 248.·13-s + 816.·14-s − 474.·15-s + 862.·16-s + 246.·17-s + 1.13e3·18-s + 31.8·19-s − 1.55e3·20-s + 1.59e3·21-s + 2.12e3·22-s + 529·23-s + 5.58e3·24-s + 625·25-s − 2.41e3·26-s − 2.38e3·27-s + 5.23e3·28-s + ⋯
L(s)  = 1  + 1.71·2-s + 1.21·3-s + 1.94·4-s − 0.447·5-s + 2.08·6-s + 0.648·7-s + 1.62·8-s + 0.481·9-s − 0.767·10-s + 0.545·11-s + 2.36·12-s − 0.408·13-s + 1.11·14-s − 0.544·15-s + 0.842·16-s + 0.206·17-s + 0.826·18-s + 0.0202·19-s − 0.870·20-s + 0.789·21-s + 0.937·22-s + 0.208·23-s + 1.97·24-s + 0.200·25-s − 0.700·26-s − 0.630·27-s + 1.26·28-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

\(L(3)\) \(\approx\) \(6.997788870\)
\(L(\frac12)\) \(\approx\) \(6.997788870\)
\(L(\frac{7}{2})\) 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 \( 1 - 9.71T + 32T^{2} \)
3 \( 1 - 18.9T + 243T^{2} \)
7 \( 1 - 84.0T + 1.68e4T^{2} \)
11 \( 1 - 219.T + 1.61e5T^{2} \)
13 \( 1 + 248.T + 3.71e5T^{2} \)
17 \( 1 - 246.T + 1.41e6T^{2} \)
19 \( 1 - 31.8T + 2.47e6T^{2} \)
29 \( 1 + 5.19e3T + 2.05e7T^{2} \)
31 \( 1 + 2.42e3T + 2.86e7T^{2} \)
37 \( 1 + 1.21e3T + 6.93e7T^{2} \)
41 \( 1 + 7.06e3T + 1.15e8T^{2} \)
43 \( 1 - 8.93e3T + 1.47e8T^{2} \)
47 \( 1 - 1.02e4T + 2.29e8T^{2} \)
53 \( 1 + 1.09e4T + 4.18e8T^{2} \)
59 \( 1 - 3.80e3T + 7.14e8T^{2} \)
61 \( 1 - 3.12e4T + 8.44e8T^{2} \)
67 \( 1 - 7.69e3T + 1.35e9T^{2} \)
71 \( 1 - 7.19e4T + 1.80e9T^{2} \)
73 \( 1 + 6.08e4T + 2.07e9T^{2} \)
79 \( 1 - 5.31e4T + 3.07e9T^{2} \)
83 \( 1 + 5.67e4T + 3.93e9T^{2} \)
89 \( 1 - 1.16e5T + 5.58e9T^{2} \)
97 \( 1 + 9.63e4T + 8.58e9T^{2} \)
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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

−12.89346843482305603664369565933, −11.89817716551529407714423288647, −11.02234350316505027022725172223, −9.294230106928530217496921293495, −8.033313033933758926220027203610, −7.00748528836821064351643873798, −5.45459681581859950234384551806, −4.19931995312552446048122796019, −3.27615950209065676843795693015, −2.01247695319561389550726348062, 2.01247695319561389550726348062, 3.27615950209065676843795693015, 4.19931995312552446048122796019, 5.45459681581859950234384551806, 7.00748528836821064351643873798, 8.033313033933758926220027203610, 9.294230106928530217496921293495, 11.02234350316505027022725172223, 11.89817716551529407714423288647, 12.89346843482305603664369565933

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