Properties

Label 2-245-1.1-c5-0-30
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  − 0.633·2-s − 19.5·3-s − 31.5·4-s − 25·5-s + 12.3·6-s + 40.2·8-s + 137.·9-s + 15.8·10-s + 76.2·11-s + 616.·12-s − 266.·13-s + 487.·15-s + 985.·16-s − 267.·17-s − 87.3·18-s + 1.76e3·19-s + 789.·20-s − 48.2·22-s + 4.41e3·23-s − 786.·24-s + 625·25-s + 168.·26-s + 2.05e3·27-s + 129.·29-s − 309.·30-s − 5.98e3·31-s − 1.91e3·32-s + ⋯
L(s)  = 1  − 0.111·2-s − 1.25·3-s − 0.987·4-s − 0.447·5-s + 0.140·6-s + 0.222·8-s + 0.567·9-s + 0.0500·10-s + 0.189·11-s + 1.23·12-s − 0.437·13-s + 0.559·15-s + 0.962·16-s − 0.224·17-s − 0.0635·18-s + 1.12·19-s + 0.441·20-s − 0.0212·22-s + 1.73·23-s − 0.278·24-s + 0.200·25-s + 0.0489·26-s + 0.541·27-s + 0.0284·29-s − 0.0626·30-s − 1.11·31-s − 0.330·32-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 \( 1 + 0.633T + 32T^{2} \)
3 \( 1 + 19.5T + 243T^{2} \)
11 \( 1 - 76.2T + 1.61e5T^{2} \)
13 \( 1 + 266.T + 3.71e5T^{2} \)
17 \( 1 + 267.T + 1.41e6T^{2} \)
19 \( 1 - 1.76e3T + 2.47e6T^{2} \)
23 \( 1 - 4.41e3T + 6.43e6T^{2} \)
29 \( 1 - 129.T + 2.05e7T^{2} \)
31 \( 1 + 5.98e3T + 2.86e7T^{2} \)
37 \( 1 - 8.90e3T + 6.93e7T^{2} \)
41 \( 1 + 9.26e3T + 1.15e8T^{2} \)
43 \( 1 + 6.78e3T + 1.47e8T^{2} \)
47 \( 1 + 2.51e4T + 2.29e8T^{2} \)
53 \( 1 + 3.06e3T + 4.18e8T^{2} \)
59 \( 1 - 2.36e4T + 7.14e8T^{2} \)
61 \( 1 + 3.22e4T + 8.44e8T^{2} \)
67 \( 1 + 3.19e4T + 1.35e9T^{2} \)
71 \( 1 - 5.95e4T + 1.80e9T^{2} \)
73 \( 1 - 2.41e4T + 2.07e9T^{2} \)
79 \( 1 - 1.07e5T + 3.07e9T^{2} \)
83 \( 1 + 5.51e3T + 3.93e9T^{2} \)
89 \( 1 + 3.97e4T + 5.58e9T^{2} \)
97 \( 1 - 1.31e5T + 8.58e9T^{2} \)
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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

−10.91713477306078661841818579353, −9.810998314153366364890863783086, −8.932894461174331474090058783438, −7.72809976403210896955453859307, −6.64134823976724354452919321843, −5.31512544355711873295994374703, −4.77962573700529298773484414890, −3.37036683996665923230889462949, −1.04369509050635340201056558031, 0, 1.04369509050635340201056558031, 3.37036683996665923230889462949, 4.77962573700529298773484414890, 5.31512544355711873295994374703, 6.64134823976724354452919321843, 7.72809976403210896955453859307, 8.932894461174331474090058783438, 9.810998314153366364890863783086, 10.91713477306078661841818579353

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