Properties

Label 2-245-1.1-c5-0-28
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  − 2.73·2-s + 28.3·3-s − 24.5·4-s − 25·5-s − 77.5·6-s + 154.·8-s + 559.·9-s + 68.4·10-s + 63.4·11-s − 694.·12-s + 868.·13-s − 708.·15-s + 361.·16-s − 1.18e3·17-s − 1.53e3·18-s − 1.53e3·19-s + 612.·20-s − 173.·22-s − 2.33e3·23-s + 4.38e3·24-s + 625·25-s − 2.37e3·26-s + 8.98e3·27-s + 8.16e3·29-s + 1.93e3·30-s + 2.36e3·31-s − 5.93e3·32-s + ⋯
L(s)  = 1  − 0.483·2-s + 1.81·3-s − 0.765·4-s − 0.447·5-s − 0.879·6-s + 0.854·8-s + 2.30·9-s + 0.216·10-s + 0.158·11-s − 1.39·12-s + 1.42·13-s − 0.812·15-s + 0.352·16-s − 0.998·17-s − 1.11·18-s − 0.974·19-s + 0.342·20-s − 0.0764·22-s − 0.921·23-s + 1.55·24-s + 0.200·25-s − 0.689·26-s + 2.37·27-s + 1.80·29-s + 0.393·30-s + 0.442·31-s − 1.02·32-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

\(L(3)\) \(\approx\) \(2.559770708\)
\(L(\frac12)\) \(\approx\) \(2.559770708\)
\(L(\frac{7}{2})\) 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 \( 1 + 2.73T + 32T^{2} \)
3 \( 1 - 28.3T + 243T^{2} \)
11 \( 1 - 63.4T + 1.61e5T^{2} \)
13 \( 1 - 868.T + 3.71e5T^{2} \)
17 \( 1 + 1.18e3T + 1.41e6T^{2} \)
19 \( 1 + 1.53e3T + 2.47e6T^{2} \)
23 \( 1 + 2.33e3T + 6.43e6T^{2} \)
29 \( 1 - 8.16e3T + 2.05e7T^{2} \)
31 \( 1 - 2.36e3T + 2.86e7T^{2} \)
37 \( 1 - 1.28e4T + 6.93e7T^{2} \)
41 \( 1 + 1.65e3T + 1.15e8T^{2} \)
43 \( 1 - 1.29e4T + 1.47e8T^{2} \)
47 \( 1 - 6.21e3T + 2.29e8T^{2} \)
53 \( 1 - 1.99e4T + 4.18e8T^{2} \)
59 \( 1 - 3.07e4T + 7.14e8T^{2} \)
61 \( 1 + 1.99e4T + 8.44e8T^{2} \)
67 \( 1 - 3.16e4T + 1.35e9T^{2} \)
71 \( 1 + 3.27e4T + 1.80e9T^{2} \)
73 \( 1 - 5.44e4T + 2.07e9T^{2} \)
79 \( 1 - 4.76e4T + 3.07e9T^{2} \)
83 \( 1 - 5.65e4T + 3.93e9T^{2} \)
89 \( 1 + 2.53e3T + 5.58e9T^{2} \)
97 \( 1 + 5.80e4T + 8.58e9T^{2} \)
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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

−10.88574458126788801555100890152, −9.949798936948476559314324594910, −8.945219934309692237807970118647, −8.463383382956362943025038399728, −7.85707616644752979464692889716, −6.51255118783850308029936439725, −4.37043760681561692996325066244, −3.84900572837529301844258641920, −2.41348004085787319856224003697, −0.993567703977594493397737669528, 0.993567703977594493397737669528, 2.41348004085787319856224003697, 3.84900572837529301844258641920, 4.37043760681561692996325066244, 6.51255118783850308029936439725, 7.85707616644752979464692889716, 8.463383382956362943025038399728, 8.945219934309692237807970118647, 9.949798936948476559314324594910, 10.88574458126788801555100890152

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