Properties

Label 2-7e2-1.1-c25-0-37
Degree $2$
Conductor $49$
Sign $1$
Analytic cond. $194.038$
Root an. cond. $13.9297$
Motivic weight $25$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 48·2-s + 1.95e5·3-s − 3.35e7·4-s + 7.41e8·5-s − 9.39e6·6-s + 3.22e9·8-s − 8.08e11·9-s − 3.56e10·10-s + 8.41e12·11-s − 6.56e12·12-s + 8.16e13·13-s + 1.45e14·15-s + 1.12e15·16-s + 2.51e15·17-s + 3.88e13·18-s + 6.08e15·19-s − 2.48e16·20-s − 4.04e14·22-s − 9.49e16·23-s + 6.30e14·24-s + 2.52e17·25-s − 3.91e15·26-s − 3.24e17·27-s − 2.71e17·29-s − 6.97e15·30-s − 4.29e18·31-s − 1.62e17·32-s + ⋯
L(s)  = 1  − 0.00828·2-s + 0.212·3-s − 0.999·4-s + 1.35·5-s − 0.00176·6-s + 0.0165·8-s − 0.954·9-s − 0.0112·10-s + 0.808·11-s − 0.212·12-s + 0.972·13-s + 0.289·15-s + 0.999·16-s + 1.04·17-s + 0.00791·18-s + 0.630·19-s − 1.35·20-s − 0.00670·22-s − 0.903·23-s + 0.00352·24-s + 0.847·25-s − 0.00805·26-s − 0.415·27-s − 0.142·29-s − 0.00239·30-s − 0.978·31-s − 0.0248·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $1$
Analytic conductor: \(194.038\)
Root analytic conductor: \(13.9297\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: $\chi_{49} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(2.817435190\)
\(L(\frac12)\) \(\approx\) \(2.817435190\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + 3 p^{4} T + p^{25} T^{2} \)
3 \( 1 - 7252 p^{3} T + p^{25} T^{2} \)
5 \( 1 - 29679594 p^{2} T + p^{25} T^{2} \)
11 \( 1 - 765410481732 p T + p^{25} T^{2} \)
13 \( 1 - 6280849641178 p T + p^{25} T^{2} \)
17 \( 1 - 148229413467534 p T + p^{25} T^{2} \)
19 \( 1 - 320108230016260 p T + p^{25} T^{2} \)
23 \( 1 + 4130229578100888 p T + p^{25} T^{2} \)
29 \( 1 + 271246959476737410 T + p^{25} T^{2} \)
31 \( 1 + 4291666067521509152 T + p^{25} T^{2} \)
37 \( 1 - 20301484446109126982 T + p^{25} T^{2} \)
41 \( 1 - \)\(18\!\cdots\!98\)\( T + p^{25} T^{2} \)
43 \( 1 - \)\(30\!\cdots\!56\)\( T + p^{25} T^{2} \)
47 \( 1 - \)\(92\!\cdots\!88\)\( T + p^{25} T^{2} \)
53 \( 1 + \)\(99\!\cdots\!54\)\( T + p^{25} T^{2} \)
59 \( 1 + \)\(13\!\cdots\!80\)\( T + p^{25} T^{2} \)
61 \( 1 + \)\(90\!\cdots\!02\)\( T + p^{25} T^{2} \)
67 \( 1 + \)\(26\!\cdots\!28\)\( T + p^{25} T^{2} \)
71 \( 1 + \)\(19\!\cdots\!48\)\( T + p^{25} T^{2} \)
73 \( 1 + \)\(42\!\cdots\!26\)\( T + p^{25} T^{2} \)
79 \( 1 + \)\(27\!\cdots\!60\)\( T + p^{25} T^{2} \)
83 \( 1 - \)\(93\!\cdots\!84\)\( T + p^{25} T^{2} \)
89 \( 1 - \)\(17\!\cdots\!30\)\( T + p^{25} T^{2} \)
97 \( 1 + \)\(28\!\cdots\!62\)\( T + p^{25} T^{2} \)
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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

−10.68355430263217237587209168375, −9.469006501290533398849073847758, −9.044182859500925686351674295385, −7.81445041417614261830458068080, −5.94170370803884137737360470950, −5.65101282186383198750872410310, −4.09295882378969403611780439503, −3.03279183837473352083927021575, −1.66546152538146782783415395064, −0.74748123164598841992158959506, 0.74748123164598841992158959506, 1.66546152538146782783415395064, 3.03279183837473352083927021575, 4.09295882378969403611780439503, 5.65101282186383198750872410310, 5.94170370803884137737360470950, 7.81445041417614261830458068080, 9.044182859500925686351674295385, 9.469006501290533398849073847758, 10.68355430263217237587209168375

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