Properties

Label 2-245-1.1-c5-0-41
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.48·2-s − 0.931·3-s − 19.8·4-s − 25·5-s + 3.24·6-s + 180.·8-s − 242.·9-s + 87.0·10-s + 286.·11-s + 18.5·12-s + 129.·13-s + 23.2·15-s + 6.61·16-s + 445.·17-s + 843.·18-s + 828.·19-s + 496.·20-s − 998.·22-s − 550.·23-s − 168.·24-s + 625·25-s − 452.·26-s + 451.·27-s + 4.84e3·29-s − 81.0·30-s + 5.80e3·31-s − 5.80e3·32-s + ⋯
L(s)  = 1  − 0.615·2-s − 0.0597·3-s − 0.620·4-s − 0.447·5-s + 0.0367·6-s + 0.997·8-s − 0.996·9-s + 0.275·10-s + 0.714·11-s + 0.0370·12-s + 0.213·13-s + 0.0267·15-s + 0.00645·16-s + 0.373·17-s + 0.613·18-s + 0.526·19-s + 0.277·20-s − 0.439·22-s − 0.216·23-s − 0.0596·24-s + 0.200·25-s − 0.131·26-s + 0.119·27-s + 1.06·29-s − 0.0164·30-s + 1.08·31-s − 1.00·32-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: $\chi_{245} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 245,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 3.48T + 32T^{2} \)
3 \( 1 + 0.931T + 243T^{2} \)
11 \( 1 - 286.T + 1.61e5T^{2} \)
13 \( 1 - 129.T + 3.71e5T^{2} \)
17 \( 1 - 445.T + 1.41e6T^{2} \)
19 \( 1 - 828.T + 2.47e6T^{2} \)
23 \( 1 + 550.T + 6.43e6T^{2} \)
29 \( 1 - 4.84e3T + 2.05e7T^{2} \)
31 \( 1 - 5.80e3T + 2.86e7T^{2} \)
37 \( 1 + 1.40e4T + 6.93e7T^{2} \)
41 \( 1 + 1.18e4T + 1.15e8T^{2} \)
43 \( 1 - 1.58e4T + 1.47e8T^{2} \)
47 \( 1 + 1.03e4T + 2.29e8T^{2} \)
53 \( 1 - 7.47e3T + 4.18e8T^{2} \)
59 \( 1 + 2.25e4T + 7.14e8T^{2} \)
61 \( 1 + 2.19e4T + 8.44e8T^{2} \)
67 \( 1 + 1.26e3T + 1.35e9T^{2} \)
71 \( 1 - 2.58e4T + 1.80e9T^{2} \)
73 \( 1 + 4.00e3T + 2.07e9T^{2} \)
79 \( 1 + 6.48e4T + 3.07e9T^{2} \)
83 \( 1 + 9.15e4T + 3.93e9T^{2} \)
89 \( 1 + 1.20e5T + 5.58e9T^{2} \)
97 \( 1 + 1.59e5T + 8.58e9T^{2} \)
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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

−10.63536449867408341366473114286, −9.679507180541610888160367424349, −8.685523828771044272052092357325, −8.136777150917029717575689308008, −6.89064120182838122905440724554, −5.55938079860846576032661275872, −4.38945859348580681424039218890, −3.16600307676482230185090638794, −1.23875696718686591450149867692, 0, 1.23875696718686591450149867692, 3.16600307676482230185090638794, 4.38945859348580681424039218890, 5.55938079860846576032661275872, 6.89064120182838122905440724554, 8.136777150917029717575689308008, 8.685523828771044272052092357325, 9.679507180541610888160367424349, 10.63536449867408341366473114286

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