Properties

Label 2-245-1.1-c5-0-36
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  + 3.66·2-s − 29.0·3-s − 18.5·4-s − 25·5-s − 106.·6-s − 185.·8-s + 601.·9-s − 91.5·10-s + 621.·11-s + 540.·12-s − 95.0·13-s + 726.·15-s − 83.0·16-s + 1.37e3·17-s + 2.20e3·18-s − 1.64e3·19-s + 464.·20-s + 2.27e3·22-s + 678.·23-s + 5.38e3·24-s + 625·25-s − 347.·26-s − 1.04e4·27-s − 6.10e3·29-s + 2.66e3·30-s − 1.45e3·31-s + 5.62e3·32-s + ⋯
L(s)  = 1  + 0.647·2-s − 1.86·3-s − 0.581·4-s − 0.447·5-s − 1.20·6-s − 1.02·8-s + 2.47·9-s − 0.289·10-s + 1.54·11-s + 1.08·12-s − 0.155·13-s + 0.833·15-s − 0.0811·16-s + 1.15·17-s + 1.60·18-s − 1.04·19-s + 0.259·20-s + 1.00·22-s + 0.267·23-s + 1.90·24-s + 0.200·25-s − 0.100·26-s − 2.75·27-s − 1.34·29-s + 0.539·30-s − 0.271·31-s + 0.970·32-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 \( 1 - 3.66T + 32T^{2} \)
3 \( 1 + 29.0T + 243T^{2} \)
11 \( 1 - 621.T + 1.61e5T^{2} \)
13 \( 1 + 95.0T + 3.71e5T^{2} \)
17 \( 1 - 1.37e3T + 1.41e6T^{2} \)
19 \( 1 + 1.64e3T + 2.47e6T^{2} \)
23 \( 1 - 678.T + 6.43e6T^{2} \)
29 \( 1 + 6.10e3T + 2.05e7T^{2} \)
31 \( 1 + 1.45e3T + 2.86e7T^{2} \)
37 \( 1 - 1.09e4T + 6.93e7T^{2} \)
41 \( 1 + 1.65e4T + 1.15e8T^{2} \)
43 \( 1 - 9.82e3T + 1.47e8T^{2} \)
47 \( 1 - 2.71e4T + 2.29e8T^{2} \)
53 \( 1 + 1.01e4T + 4.18e8T^{2} \)
59 \( 1 + 2.32e4T + 7.14e8T^{2} \)
61 \( 1 - 3.04e4T + 8.44e8T^{2} \)
67 \( 1 + 1.43e4T + 1.35e9T^{2} \)
71 \( 1 + 5.64e4T + 1.80e9T^{2} \)
73 \( 1 + 2.05e4T + 2.07e9T^{2} \)
79 \( 1 + 7.50e4T + 3.07e9T^{2} \)
83 \( 1 + 4.48e4T + 3.93e9T^{2} \)
89 \( 1 - 1.41e5T + 5.58e9T^{2} \)
97 \( 1 - 3.31e4T + 8.58e9T^{2} \)
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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

−11.09930000661200983602824237762, −9.989290652532929899942391935050, −8.997019126390413131124588466820, −7.37790361701986341433545675869, −6.28459243135349957222240395736, −5.59641174970581705289536001172, −4.49279256800255873452724451578, −3.79773350985293044677337326940, −1.16335303497641201169324673857, 0, 1.16335303497641201169324673857, 3.79773350985293044677337326940, 4.49279256800255873452724451578, 5.59641174970581705289536001172, 6.28459243135349957222240395736, 7.37790361701986341433545675869, 8.997019126390413131124588466820, 9.989290652532929899942391935050, 11.09930000661200983602824237762

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