Properties

Label 2-7e2-1.1-c25-0-16
Degree $2$
Conductor $49$
Sign $1$
Analytic cond. $194.038$
Root an. cond. $13.9297$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.60e3·2-s − 4.93e5·3-s − 2.67e7·4-s + 3.09e8·5-s + 1.28e9·6-s + 1.57e11·8-s − 6.03e11·9-s − 8.05e11·10-s + 9.78e12·11-s + 1.32e13·12-s − 1.58e14·13-s − 1.52e14·15-s + 4.89e14·16-s + 2.90e15·17-s + 1.57e15·18-s − 1.67e16·19-s − 8.28e15·20-s − 2.54e16·22-s + 1.60e17·23-s − 7.75e16·24-s − 2.02e17·25-s + 4.12e17·26-s + 7.16e17·27-s + 1.12e18·29-s + 3.97e17·30-s − 1.46e17·31-s − 6.54e18·32-s + ⋯
L(s)  = 1  − 0.449·2-s − 0.536·3-s − 0.797·4-s + 0.566·5-s + 0.241·6-s + 0.808·8-s − 0.712·9-s − 0.254·10-s + 0.939·11-s + 0.428·12-s − 1.88·13-s − 0.304·15-s + 0.434·16-s + 1.21·17-s + 0.320·18-s − 1.73·19-s − 0.452·20-s − 0.422·22-s + 1.52·23-s − 0.433·24-s − 0.678·25-s + 0.847·26-s + 0.918·27-s + 0.588·29-s + 0.136·30-s − 0.0333·31-s − 1.00·32-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(0.6704707929\)
\(L(\frac12)\) \(\approx\) \(0.6704707929\)
\(L(\frac{27}{2})\) 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 \( 1 + 2.60e3T + 3.35e7T^{2} \)
3 \( 1 + 4.93e5T + 8.47e11T^{2} \)
5 \( 1 - 3.09e8T + 2.98e17T^{2} \)
11 \( 1 - 9.78e12T + 1.08e26T^{2} \)
13 \( 1 + 1.58e14T + 7.05e27T^{2} \)
17 \( 1 - 2.90e15T + 5.77e30T^{2} \)
19 \( 1 + 1.67e16T + 9.30e31T^{2} \)
23 \( 1 - 1.60e17T + 1.10e34T^{2} \)
29 \( 1 - 1.12e18T + 3.63e36T^{2} \)
31 \( 1 + 1.46e17T + 1.92e37T^{2} \)
37 \( 1 + 2.78e18T + 1.60e39T^{2} \)
41 \( 1 - 2.26e20T + 2.08e40T^{2} \)
43 \( 1 + 1.58e20T + 6.86e40T^{2} \)
47 \( 1 + 4.78e20T + 6.34e41T^{2} \)
53 \( 1 + 4.38e21T + 1.27e43T^{2} \)
59 \( 1 + 1.71e22T + 1.86e44T^{2} \)
61 \( 1 - 1.89e21T + 4.29e44T^{2} \)
67 \( 1 + 6.63e22T + 4.48e45T^{2} \)
71 \( 1 + 2.36e22T + 1.91e46T^{2} \)
73 \( 1 + 3.17e23T + 3.82e46T^{2} \)
79 \( 1 - 7.85e23T + 2.75e47T^{2} \)
83 \( 1 + 4.07e23T + 9.48e47T^{2} \)
89 \( 1 + 3.03e24T + 5.42e48T^{2} \)
97 \( 1 - 7.83e24T + 4.66e49T^{2} \)
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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

−10.67677153431658425652090406983, −9.708530644251513412014855882897, −8.926729413295858086945366972485, −7.69960999939978282222924461714, −6.37868636081653304185015914500, −5.26480935468962085650179268148, −4.42200399847170676809115214891, −2.86741831962622919659657252060, −1.52195421540971887158138478698, −0.39699950675252461574221265553, 0.39699950675252461574221265553, 1.52195421540971887158138478698, 2.86741831962622919659657252060, 4.42200399847170676809115214891, 5.26480935468962085650179268148, 6.37868636081653304185015914500, 7.69960999939978282222924461714, 8.926729413295858086945366972485, 9.708530644251513412014855882897, 10.67677153431658425652090406983

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