Properties

Label 2-1-1.1-c49-0-0
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $15.2066$
Root an. cond. $3.89956$
Motivic weight $49$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.54e7·2-s − 8.64e11·3-s + 8.31e13·4-s + 1.67e17·5-s + 2.19e19·6-s − 3.42e20·7-s + 1.21e22·8-s + 5.08e23·9-s − 4.25e24·10-s − 2.47e24·11-s − 7.18e25·12-s − 1.86e26·13-s + 8.71e27·14-s − 1.44e29·15-s − 3.56e29·16-s + 8.08e29·17-s − 1.29e31·18-s − 1.31e31·19-s + 1.39e31·20-s + 2.96e32·21-s + 6.28e31·22-s + 3.96e33·23-s − 1.05e34·24-s + 1.02e34·25-s + 4.74e33·26-s − 2.32e35·27-s − 2.84e34·28-s + ⋯
L(s)  = 1  − 1.07·2-s − 1.76·3-s + 0.147·4-s + 1.25·5-s + 1.89·6-s − 0.676·7-s + 0.913·8-s + 2.12·9-s − 1.34·10-s − 0.0756·11-s − 0.261·12-s − 0.0952·13-s + 0.724·14-s − 2.22·15-s − 1.12·16-s + 0.577·17-s − 2.27·18-s − 0.616·19-s + 0.185·20-s + 1.19·21-s + 0.0810·22-s + 1.72·23-s − 1.61·24-s + 0.577·25-s + 0.102·26-s − 1.98·27-s − 0.0998·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(50-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+49/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(15.2066\)
Root analytic conductor: \(3.89956\)
Motivic weight: \(49\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :49/2),\ -1)\)

Particular Values

\(L(25)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{51}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 2.54e7T + 5.62e14T^{2} \)
3 \( 1 + 8.64e11T + 2.39e23T^{2} \)
5 \( 1 - 1.67e17T + 1.77e34T^{2} \)
7 \( 1 + 3.42e20T + 2.56e41T^{2} \)
11 \( 1 + 2.47e24T + 1.06e51T^{2} \)
13 \( 1 + 1.86e26T + 3.83e54T^{2} \)
17 \( 1 - 8.08e29T + 1.95e60T^{2} \)
19 \( 1 + 1.31e31T + 4.55e62T^{2} \)
23 \( 1 - 3.96e33T + 5.30e66T^{2} \)
29 \( 1 - 4.73e35T + 4.54e71T^{2} \)
31 \( 1 + 5.52e36T + 1.19e73T^{2} \)
37 \( 1 - 6.02e37T + 6.94e76T^{2} \)
41 \( 1 + 4.00e39T + 1.06e79T^{2} \)
43 \( 1 + 1.57e39T + 1.09e80T^{2} \)
47 \( 1 + 3.77e40T + 8.56e81T^{2} \)
53 \( 1 + 7.38e41T + 3.08e84T^{2} \)
59 \( 1 + 3.19e43T + 5.91e86T^{2} \)
61 \( 1 - 2.48e43T + 3.02e87T^{2} \)
67 \( 1 - 5.76e44T + 3.00e89T^{2} \)
71 \( 1 + 1.66e45T + 5.14e90T^{2} \)
73 \( 1 - 4.80e45T + 2.00e91T^{2} \)
79 \( 1 + 4.21e45T + 9.63e92T^{2} \)
83 \( 1 + 1.20e47T + 1.08e94T^{2} \)
89 \( 1 - 8.70e47T + 3.31e95T^{2} \)
97 \( 1 + 7.30e48T + 2.24e97T^{2} \)
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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

−18.66258266906742905187920213838, −17.42533108564586364924787074572, −16.59357600966871094712988073268, −12.97200104867324834434759329215, −10.72193136381819341197236094422, −9.584881447441338852328025424151, −6.68829227134675823405056170950, −5.19645735666897088606660426356, −1.36617840071946262905158063342, 0, 1.36617840071946262905158063342, 5.19645735666897088606660426356, 6.68829227134675823405056170950, 9.584881447441338852328025424151, 10.72193136381819341197236094422, 12.97200104867324834434759329215, 16.59357600966871094712988073268, 17.42533108564586364924787074572, 18.66258266906742905187920213838

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