Properties

Label 2-1-1.1-c47-0-3
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $13.9907$
Root an. cond. $3.74042$
Motivic weight $47$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30e7·2-s + 1.28e11·3-s + 3.91e14·4-s − 1.15e16·5-s + 2.96e18·6-s + 1.54e19·7-s + 5.79e21·8-s − 1.00e22·9-s − 2.65e23·10-s − 3.19e24·11-s + 5.03e25·12-s + 1.57e25·13-s + 3.55e26·14-s − 1.47e27·15-s + 7.86e28·16-s − 3.42e28·17-s − 2.32e29·18-s + 1.00e30·19-s − 4.51e30·20-s + 1.98e30·21-s − 7.37e31·22-s + 5.17e31·23-s + 7.44e32·24-s − 5.78e32·25-s + 3.64e32·26-s − 4.71e33·27-s + 6.04e33·28-s + ⋯
L(s)  = 1  + 1.94·2-s + 0.788·3-s + 2.78·4-s − 0.431·5-s + 1.53·6-s + 0.212·7-s + 3.47·8-s − 0.378·9-s − 0.839·10-s − 1.07·11-s + 2.19·12-s + 0.104·13-s + 0.414·14-s − 0.340·15-s + 3.97·16-s − 0.416·17-s − 0.737·18-s + 0.889·19-s − 1.20·20-s + 0.167·21-s − 2.09·22-s + 0.517·23-s + 2.73·24-s − 0.813·25-s + 0.203·26-s − 1.08·27-s + 0.593·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(48-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+47/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(13.9907\)
Root analytic conductor: \(3.74042\)
Motivic weight: \(47\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :47/2),\ 1)\)

Particular Values

\(L(24)\) \(\approx\) \(6.573291442\)
\(L(\frac12)\) \(\approx\) \(6.573291442\)
\(L(\frac{49}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.30e7T + 1.40e14T^{2} \)
3 \( 1 - 1.28e11T + 2.65e22T^{2} \)
5 \( 1 + 1.15e16T + 7.10e32T^{2} \)
7 \( 1 - 1.54e19T + 5.24e39T^{2} \)
11 \( 1 + 3.19e24T + 8.81e48T^{2} \)
13 \( 1 - 1.57e25T + 2.26e52T^{2} \)
17 \( 1 + 3.42e28T + 6.77e57T^{2} \)
19 \( 1 - 1.00e30T + 1.26e60T^{2} \)
23 \( 1 - 5.17e31T + 1.00e64T^{2} \)
29 \( 1 + 6.90e33T + 5.40e68T^{2} \)
31 \( 1 + 1.79e35T + 1.24e70T^{2} \)
37 \( 1 + 2.68e36T + 5.07e73T^{2} \)
41 \( 1 - 1.17e38T + 6.32e75T^{2} \)
43 \( 1 - 1.58e38T + 5.92e76T^{2} \)
47 \( 1 + 1.34e39T + 3.87e78T^{2} \)
53 \( 1 - 4.68e40T + 1.09e81T^{2} \)
59 \( 1 - 3.84e41T + 1.69e83T^{2} \)
61 \( 1 - 4.37e41T + 8.13e83T^{2} \)
67 \( 1 - 4.64e42T + 6.69e85T^{2} \)
71 \( 1 - 1.91e43T + 1.02e87T^{2} \)
73 \( 1 - 8.25e43T + 3.76e87T^{2} \)
79 \( 1 + 5.45e44T + 1.54e89T^{2} \)
83 \( 1 + 1.02e45T + 1.57e90T^{2} \)
89 \( 1 - 1.06e46T + 4.18e91T^{2} \)
97 \( 1 - 3.72e46T + 2.38e93T^{2} \)
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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

−21.07516266632649335123314739269, −19.88095368143506660305027662922, −15.82196180206301809420558655544, −14.48598525075133139211032382979, −13.11442815552733184083047136647, −11.28244498175268667193628020180, −7.59247495885116737306597296132, −5.40006152176154725533900714111, −3.64203140782896131177921063585, −2.34778698908042292893419555331, 2.34778698908042292893419555331, 3.64203140782896131177921063585, 5.40006152176154725533900714111, 7.59247495885116737306597296132, 11.28244498175268667193628020180, 13.11442815552733184083047136647, 14.48598525075133139211032382979, 15.82196180206301809420558655544, 19.88095368143506660305027662922, 21.07516266632649335123314739269

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