Properties

Label 2-245-1.1-c5-0-64
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  + 10.3·2-s − 10.7·3-s + 74.7·4-s − 25·5-s − 111.·6-s + 442.·8-s − 126.·9-s − 258.·10-s − 262.·11-s − 806.·12-s − 688.·13-s + 269.·15-s + 2.17e3·16-s − 1.67e3·17-s − 1.31e3·18-s − 925.·19-s − 1.86e3·20-s − 2.71e3·22-s + 4.60e3·23-s − 4.76e3·24-s + 625·25-s − 7.11e3·26-s + 3.98e3·27-s − 3.41e3·29-s + 2.78e3·30-s + 5.53e3·31-s + 8.34e3·32-s + ⋯
L(s)  = 1  + 1.82·2-s − 0.691·3-s + 2.33·4-s − 0.447·5-s − 1.26·6-s + 2.44·8-s − 0.521·9-s − 0.816·10-s − 0.655·11-s − 1.61·12-s − 1.13·13-s + 0.309·15-s + 2.12·16-s − 1.40·17-s − 0.953·18-s − 0.587·19-s − 1.04·20-s − 1.19·22-s + 1.81·23-s − 1.68·24-s + 0.200·25-s − 2.06·26-s + 1.05·27-s − 0.753·29-s + 0.564·30-s + 1.03·31-s + 1.44·32-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 \( 1 - 10.3T + 32T^{2} \)
3 \( 1 + 10.7T + 243T^{2} \)
11 \( 1 + 262.T + 1.61e5T^{2} \)
13 \( 1 + 688.T + 3.71e5T^{2} \)
17 \( 1 + 1.67e3T + 1.41e6T^{2} \)
19 \( 1 + 925.T + 2.47e6T^{2} \)
23 \( 1 - 4.60e3T + 6.43e6T^{2} \)
29 \( 1 + 3.41e3T + 2.05e7T^{2} \)
31 \( 1 - 5.53e3T + 2.86e7T^{2} \)
37 \( 1 + 1.49e4T + 6.93e7T^{2} \)
41 \( 1 + 1.49e4T + 1.15e8T^{2} \)
43 \( 1 + 1.96e4T + 1.47e8T^{2} \)
47 \( 1 - 1.12e4T + 2.29e8T^{2} \)
53 \( 1 + 2.13e4T + 4.18e8T^{2} \)
59 \( 1 - 3.47e3T + 7.14e8T^{2} \)
61 \( 1 - 2.23e4T + 8.44e8T^{2} \)
67 \( 1 - 4.49e4T + 1.35e9T^{2} \)
71 \( 1 - 1.09e4T + 1.80e9T^{2} \)
73 \( 1 - 9.20e3T + 2.07e9T^{2} \)
79 \( 1 + 3.35e3T + 3.07e9T^{2} \)
83 \( 1 + 4.55e4T + 3.93e9T^{2} \)
89 \( 1 - 5.28e4T + 5.58e9T^{2} \)
97 \( 1 - 1.64e5T + 8.58e9T^{2} \)
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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

−11.21549095824338804720739150306, −10.41957847493319476436960878432, −8.599613728069086516030396174759, −7.12626539862198594140891431569, −6.49284686452283808900517150745, −5.11733264437658635675443882789, −4.83567140378609100515879814365, −3.34017073998184507281108673101, −2.27312931224302467961178707324, 0, 2.27312931224302467961178707324, 3.34017073998184507281108673101, 4.83567140378609100515879814365, 5.11733264437658635675443882789, 6.49284686452283808900517150745, 7.12626539862198594140891431569, 8.599613728069086516030396174759, 10.41957847493319476436960878432, 11.21549095824338804720739150306

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