Properties

Label 2-7e2-1.1-c25-0-66
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.61e3·2-s + 1.62e6·3-s + 5.89e7·4-s − 7.42e8·5-s − 1.56e10·6-s − 2.44e11·8-s + 1.78e12·9-s + 7.13e12·10-s + 1.28e13·11-s + 9.56e13·12-s + 4.93e13·13-s − 1.20e15·15-s + 3.69e14·16-s + 2.28e14·17-s − 1.71e16·18-s + 2.89e15·19-s − 4.37e16·20-s − 1.23e17·22-s − 1.62e17·23-s − 3.96e17·24-s + 2.52e17·25-s − 4.74e17·26-s + 1.52e18·27-s − 1.10e18·29-s + 1.15e19·30-s − 3.63e18·31-s + 4.63e18·32-s + ⋯
L(s)  = 1  − 1.66·2-s + 1.76·3-s + 1.75·4-s − 1.35·5-s − 2.92·6-s − 1.25·8-s + 2.10·9-s + 2.25·10-s + 1.23·11-s + 3.09·12-s + 0.587·13-s − 2.39·15-s + 0.328·16-s + 0.0952·17-s − 3.49·18-s + 0.299·19-s − 2.38·20-s − 2.04·22-s − 1.54·23-s − 2.21·24-s + 0.848·25-s − 0.974·26-s + 1.95·27-s − 0.579·29-s + 3.97·30-s − 0.829·31-s + 0.710·32-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: \(1\)
Selberg data: \((2,\ 49,\ (\ :25/2),\ -1)\)

Particular Values

\(L(13)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{27}{2})\) 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 \( 1 + 9.61e3T + 3.35e7T^{2} \)
3 \( 1 - 1.62e6T + 8.47e11T^{2} \)
5 \( 1 + 7.42e8T + 2.98e17T^{2} \)
11 \( 1 - 1.28e13T + 1.08e26T^{2} \)
13 \( 1 - 4.93e13T + 7.05e27T^{2} \)
17 \( 1 - 2.28e14T + 5.77e30T^{2} \)
19 \( 1 - 2.89e15T + 9.30e31T^{2} \)
23 \( 1 + 1.62e17T + 1.10e34T^{2} \)
29 \( 1 + 1.10e18T + 3.63e36T^{2} \)
31 \( 1 + 3.63e18T + 1.92e37T^{2} \)
37 \( 1 + 2.12e19T + 1.60e39T^{2} \)
41 \( 1 + 1.90e20T + 2.08e40T^{2} \)
43 \( 1 - 2.97e20T + 6.86e40T^{2} \)
47 \( 1 + 4.41e20T + 6.34e41T^{2} \)
53 \( 1 + 4.75e21T + 1.27e43T^{2} \)
59 \( 1 - 1.06e22T + 1.86e44T^{2} \)
61 \( 1 - 1.37e22T + 4.29e44T^{2} \)
67 \( 1 + 2.33e22T + 4.48e45T^{2} \)
71 \( 1 - 2.86e22T + 1.91e46T^{2} \)
73 \( 1 - 1.20e23T + 3.82e46T^{2} \)
79 \( 1 - 4.94e23T + 2.75e47T^{2} \)
83 \( 1 - 1.05e24T + 9.48e47T^{2} \)
89 \( 1 - 1.33e23T + 5.42e48T^{2} \)
97 \( 1 + 1.29e25T + 4.66e49T^{2} \)
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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

−9.817738902263486343502198795838, −9.023518174476500481800106373216, −8.250063895646939661078485435091, −7.68739269302256244565273396638, −6.74472348136016962244722583482, −4.03111406820256402183470558165, −3.44073914105820670598992340045, −2.05076370728922134848864303253, −1.23792027769220174887301191947, 0, 1.23792027769220174887301191947, 2.05076370728922134848864303253, 3.44073914105820670598992340045, 4.03111406820256402183470558165, 6.74472348136016962244722583482, 7.68739269302256244565273396638, 8.250063895646939661078485435091, 9.023518174476500481800106373216, 9.817738902263486343502198795838

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