Properties

Label 2-245-1.1-c3-0-31
Degree $2$
Conductor $245$
Sign $1$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
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  + 4.46·2-s + 9.80·3-s + 11.9·4-s − 5·5-s + 43.8·6-s + 17.6·8-s + 69.2·9-s − 22.3·10-s − 56.5·11-s + 117.·12-s + 40.9·13-s − 49.0·15-s − 16.6·16-s + 2.18·17-s + 309.·18-s + 16.4·19-s − 59.7·20-s − 252.·22-s − 155.·23-s + 173.·24-s + 25·25-s + 183.·26-s + 414.·27-s − 6.26·29-s − 219.·30-s + 168.·31-s − 215.·32-s + ⋯
L(s)  = 1  + 1.57·2-s + 1.88·3-s + 1.49·4-s − 0.447·5-s + 2.98·6-s + 0.781·8-s + 2.56·9-s − 0.706·10-s − 1.54·11-s + 2.82·12-s + 0.873·13-s − 0.844·15-s − 0.260·16-s + 0.0312·17-s + 4.04·18-s + 0.198·19-s − 0.668·20-s − 2.44·22-s − 1.40·23-s + 1.47·24-s + 0.200·25-s + 1.38·26-s + 2.95·27-s − 0.0400·29-s − 1.33·30-s + 0.977·31-s − 1.19·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.786901902\)
\(L(\frac12)\) \(\approx\) \(6.786901902\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
7 \( 1 \)
good2 \( 1 - 4.46T + 8T^{2} \)
3 \( 1 - 9.80T + 27T^{2} \)
11 \( 1 + 56.5T + 1.33e3T^{2} \)
13 \( 1 - 40.9T + 2.19e3T^{2} \)
17 \( 1 - 2.18T + 4.91e3T^{2} \)
19 \( 1 - 16.4T + 6.85e3T^{2} \)
23 \( 1 + 155.T + 1.21e4T^{2} \)
29 \( 1 + 6.26T + 2.43e4T^{2} \)
31 \( 1 - 168.T + 2.97e4T^{2} \)
37 \( 1 + 37.1T + 5.06e4T^{2} \)
41 \( 1 + 266.T + 6.89e4T^{2} \)
43 \( 1 + 14.6T + 7.95e4T^{2} \)
47 \( 1 + 169.T + 1.03e5T^{2} \)
53 \( 1 + 151.T + 1.48e5T^{2} \)
59 \( 1 + 234.T + 2.05e5T^{2} \)
61 \( 1 + 242.T + 2.26e5T^{2} \)
67 \( 1 - 820.T + 3.00e5T^{2} \)
71 \( 1 - 961.T + 3.57e5T^{2} \)
73 \( 1 - 934.T + 3.89e5T^{2} \)
79 \( 1 - 300.T + 4.93e5T^{2} \)
83 \( 1 - 1.08e3T + 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 - 752.T + 9.12e5T^{2} \)
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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

−12.16411677403737909495104306173, −10.80055542375259916580304044283, −9.711465170946422691362550975979, −8.353364261499241539803393341360, −7.82765693874826438970777311765, −6.56411547477754575625049557055, −5.04380128049639869261488655691, −3.93172804780944131526303425469, −3.16501718602807753075128091919, −2.16401220417760296020271095160, 2.16401220417760296020271095160, 3.16501718602807753075128091919, 3.93172804780944131526303425469, 5.04380128049639869261488655691, 6.56411547477754575625049557055, 7.82765693874826438970777311765, 8.353364261499241539803393341360, 9.711465170946422691362550975979, 10.80055542375259916580304044283, 12.16411677403737909495104306173

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