Properties

Label 2-7e2-1.1-c25-0-21
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  + 9.69e3·2-s − 8.94e5·3-s + 6.04e7·4-s − 4.19e8·5-s − 8.67e9·6-s + 2.61e11·8-s − 4.78e10·9-s − 4.07e12·10-s + 1.33e12·11-s − 5.40e13·12-s − 6.92e13·13-s + 3.75e14·15-s + 5.02e14·16-s − 3.99e15·17-s − 4.63e14·18-s − 3.21e15·19-s − 2.53e16·20-s + 1.29e16·22-s + 6.09e16·23-s − 2.33e17·24-s − 1.21e17·25-s − 6.71e17·26-s + 8.00e17·27-s + 3.08e18·29-s + 3.64e18·30-s + 6.57e18·31-s − 3.88e18·32-s + ⋯
L(s)  = 1  + 1.67·2-s − 0.971·3-s + 1.80·4-s − 0.769·5-s − 1.62·6-s + 1.34·8-s − 0.0564·9-s − 1.28·10-s + 0.128·11-s − 1.75·12-s − 0.824·13-s + 0.747·15-s + 0.446·16-s − 1.66·17-s − 0.0944·18-s − 0.333·19-s − 1.38·20-s + 0.214·22-s + 0.580·23-s − 1.30·24-s − 0.408·25-s − 1.37·26-s + 1.02·27-s + 1.61·29-s + 1.25·30-s + 1.50·31-s − 0.596·32-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

\(L(13)\) \(\approx\) \(2.212057929\)
\(L(\frac12)\) \(\approx\) \(2.212057929\)
\(L(\frac{27}{2})\) 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 \( 1 - 9.69e3T + 3.35e7T^{2} \)
3 \( 1 + 8.94e5T + 8.47e11T^{2} \)
5 \( 1 + 4.19e8T + 2.98e17T^{2} \)
11 \( 1 - 1.33e12T + 1.08e26T^{2} \)
13 \( 1 + 6.92e13T + 7.05e27T^{2} \)
17 \( 1 + 3.99e15T + 5.77e30T^{2} \)
19 \( 1 + 3.21e15T + 9.30e31T^{2} \)
23 \( 1 - 6.09e16T + 1.10e34T^{2} \)
29 \( 1 - 3.08e18T + 3.63e36T^{2} \)
31 \( 1 - 6.57e18T + 1.92e37T^{2} \)
37 \( 1 - 3.76e18T + 1.60e39T^{2} \)
41 \( 1 + 2.38e20T + 2.08e40T^{2} \)
43 \( 1 + 2.96e20T + 6.86e40T^{2} \)
47 \( 1 + 8.07e20T + 6.34e41T^{2} \)
53 \( 1 - 4.02e21T + 1.27e43T^{2} \)
59 \( 1 + 6.81e21T + 1.86e44T^{2} \)
61 \( 1 - 1.82e21T + 4.29e44T^{2} \)
67 \( 1 - 1.02e23T + 4.48e45T^{2} \)
71 \( 1 + 6.11e22T + 1.91e46T^{2} \)
73 \( 1 - 2.19e21T + 3.82e46T^{2} \)
79 \( 1 + 1.84e23T + 2.75e47T^{2} \)
83 \( 1 - 1.18e24T + 9.48e47T^{2} \)
89 \( 1 - 2.36e24T + 5.42e48T^{2} \)
97 \( 1 - 5.26e24T + 4.66e49T^{2} \)
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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

−11.63726754503515436737902391483, −10.39765315567138527675534856094, −8.467634837924676794400847585553, −6.86155293528280022642402563208, −6.31414783520212630008759120469, −4.92298945579584985241521823621, −4.57664832362640297178816426382, −3.27987036343785529466418275000, −2.22152589158804324214045879382, −0.48071225608662265303255322401, 0.48071225608662265303255322401, 2.22152589158804324214045879382, 3.27987036343785529466418275000, 4.57664832362640297178816426382, 4.92298945579584985241521823621, 6.31414783520212630008759120469, 6.86155293528280022642402563208, 8.467634837924676794400847585553, 10.39765315567138527675534856094, 11.63726754503515436737902391483

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