Properties

Label 2-245-1.1-c5-0-10
Degree $2$
Conductor $245$
Sign $1$
Analytic cond. $39.2940$
Root an. cond. $6.26849$
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  − 7.03·2-s + 10.0·3-s + 17.4·4-s − 25·5-s − 70.7·6-s + 102.·8-s − 141.·9-s + 175.·10-s − 125.·11-s + 175.·12-s + 40.4·13-s − 251.·15-s − 1.27e3·16-s + 1.14e3·17-s + 996.·18-s − 2.09e3·19-s − 435.·20-s + 882.·22-s + 1.22e3·23-s + 1.03e3·24-s + 625·25-s − 284.·26-s − 3.87e3·27-s + 3.03e3·29-s + 1.76e3·30-s + 2.11e3·31-s + 5.70e3·32-s + ⋯
L(s)  = 1  − 1.24·2-s + 0.645·3-s + 0.544·4-s − 0.447·5-s − 0.802·6-s + 0.565·8-s − 0.583·9-s + 0.555·10-s − 0.312·11-s + 0.351·12-s + 0.0663·13-s − 0.288·15-s − 1.24·16-s + 0.963·17-s + 0.724·18-s − 1.33·19-s − 0.243·20-s + 0.388·22-s + 0.481·23-s + 0.365·24-s + 0.200·25-s − 0.0824·26-s − 1.02·27-s + 0.669·29-s + 0.358·30-s + 0.395·31-s + 0.985·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(39.2940\)
Root analytic conductor: \(6.26849\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8415069719\)
\(L(\frac12)\) \(\approx\) \(0.8415069719\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25T \)
7 \( 1 \)
good2 \( 1 + 7.03T + 32T^{2} \)
3 \( 1 - 10.0T + 243T^{2} \)
11 \( 1 + 125.T + 1.61e5T^{2} \)
13 \( 1 - 40.4T + 3.71e5T^{2} \)
17 \( 1 - 1.14e3T + 1.41e6T^{2} \)
19 \( 1 + 2.09e3T + 2.47e6T^{2} \)
23 \( 1 - 1.22e3T + 6.43e6T^{2} \)
29 \( 1 - 3.03e3T + 2.05e7T^{2} \)
31 \( 1 - 2.11e3T + 2.86e7T^{2} \)
37 \( 1 + 1.02e3T + 6.93e7T^{2} \)
41 \( 1 + 2.13e4T + 1.15e8T^{2} \)
43 \( 1 - 6.91e3T + 1.47e8T^{2} \)
47 \( 1 - 2.67e3T + 2.29e8T^{2} \)
53 \( 1 + 2.23e4T + 4.18e8T^{2} \)
59 \( 1 - 2.11e4T + 7.14e8T^{2} \)
61 \( 1 - 1.96e4T + 8.44e8T^{2} \)
67 \( 1 - 1.93e4T + 1.35e9T^{2} \)
71 \( 1 + 3.89e3T + 1.80e9T^{2} \)
73 \( 1 - 5.36e4T + 2.07e9T^{2} \)
79 \( 1 - 9.64e4T + 3.07e9T^{2} \)
83 \( 1 - 1.03e5T + 3.93e9T^{2} \)
89 \( 1 - 7.87e4T + 5.58e9T^{2} \)
97 \( 1 - 1.29e5T + 8.58e9T^{2} \)
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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

−10.90552149129428882748885349421, −10.12679781626748296713314959247, −9.089766749004894165996052417668, −8.340056330205293164137542904498, −7.80306376684413280298386066418, −6.56908449043767954318890210697, −4.95674283403296754428735764283, −3.50482403757672202163604678302, −2.15356661184456300390551076150, −0.62037561394133625500276782714, 0.62037561394133625500276782714, 2.15356661184456300390551076150, 3.50482403757672202163604678302, 4.95674283403296754428735764283, 6.56908449043767954318890210697, 7.80306376684413280298386066418, 8.340056330205293164137542904498, 9.089766749004894165996052417668, 10.12679781626748296713314959247, 10.90552149129428882748885349421

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