Properties

Label 2-7e2-1.1-c5-0-7
Degree $2$
Conductor $49$
Sign $-1$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
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  − 5.08·2-s + 2.08·3-s − 6.16·4-s + 41.8·5-s − 10.5·6-s + 193.·8-s − 238.·9-s − 212.·10-s + 72.0·11-s − 12.8·12-s − 632.·13-s + 87.1·15-s − 788.·16-s − 1.97e3·17-s + 1.21e3·18-s − 1.86e3·19-s − 257.·20-s − 366.·22-s + 413.·23-s + 404.·24-s − 1.37e3·25-s + 3.21e3·26-s − 1.00e3·27-s + 731.·29-s − 442.·30-s + 6.12e3·31-s − 2.19e3·32-s + ⋯
L(s)  = 1  − 0.898·2-s + 0.133·3-s − 0.192·4-s + 0.748·5-s − 0.120·6-s + 1.07·8-s − 0.982·9-s − 0.672·10-s + 0.179·11-s − 0.0257·12-s − 1.03·13-s + 0.0999·15-s − 0.770·16-s − 1.65·17-s + 0.882·18-s − 1.18·19-s − 0.144·20-s − 0.161·22-s + 0.163·23-s + 0.143·24-s − 0.440·25-s + 0.932·26-s − 0.264·27-s + 0.161·29-s − 0.0898·30-s + 1.14·31-s − 0.379·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-1$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -1)\)

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)$
bad7 \( 1 \)
good2 \( 1 + 5.08T + 32T^{2} \)
3 \( 1 - 2.08T + 243T^{2} \)
5 \( 1 - 41.8T + 3.12e3T^{2} \)
11 \( 1 - 72.0T + 1.61e5T^{2} \)
13 \( 1 + 632.T + 3.71e5T^{2} \)
17 \( 1 + 1.97e3T + 1.41e6T^{2} \)
19 \( 1 + 1.86e3T + 2.47e6T^{2} \)
23 \( 1 - 413.T + 6.43e6T^{2} \)
29 \( 1 - 731.T + 2.05e7T^{2} \)
31 \( 1 - 6.12e3T + 2.86e7T^{2} \)
37 \( 1 - 1.03e4T + 6.93e7T^{2} \)
41 \( 1 + 3.52e3T + 1.15e8T^{2} \)
43 \( 1 + 1.45e4T + 1.47e8T^{2} \)
47 \( 1 + 2.14e4T + 2.29e8T^{2} \)
53 \( 1 - 1.25e4T + 4.18e8T^{2} \)
59 \( 1 + 3.61e4T + 7.14e8T^{2} \)
61 \( 1 - 4.02e3T + 8.44e8T^{2} \)
67 \( 1 - 1.55e4T + 1.35e9T^{2} \)
71 \( 1 - 1.21e4T + 1.80e9T^{2} \)
73 \( 1 - 1.95e4T + 2.07e9T^{2} \)
79 \( 1 - 3.60e4T + 3.07e9T^{2} \)
83 \( 1 + 2.45e4T + 3.93e9T^{2} \)
89 \( 1 + 7.02e4T + 5.58e9T^{2} \)
97 \( 1 - 1.05e5T + 8.58e9T^{2} \)
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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

−13.97707505916677750572864290375, −13.07045261790796160774708673764, −11.37549949434715106644384913737, −10.12561173926461168098254101013, −9.135193729930420440753916975988, −8.180280829456720905351442699079, −6.47594150688748095783940565143, −4.70786807799000360944408023498, −2.21371360549040054329945564287, 0, 2.21371360549040054329945564287, 4.70786807799000360944408023498, 6.47594150688748095783940565143, 8.180280829456720905351442699079, 9.135193729930420440753916975988, 10.12561173926461168098254101013, 11.37549949434715106644384913737, 13.07045261790796160774708673764, 13.97707505916677750572864290375

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