Properties

Label 2-1-1.1-c51-0-1
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $16.4731$
Root an. cond. $4.05871$
Motivic weight $51$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.71e7·2-s + 6.49e11·3-s + 5.33e15·4-s + 8.70e17·5-s − 5.65e19·6-s + 3.14e21·7-s − 2.68e23·8-s − 1.73e24·9-s − 7.58e25·10-s + 4.69e26·11-s + 3.46e27·12-s + 9.04e27·13-s − 2.73e29·14-s + 5.65e29·15-s + 1.13e31·16-s − 1.94e30·17-s + 1.50e32·18-s − 2.88e32·19-s + 4.64e33·20-s + 2.04e33·21-s − 4.09e34·22-s + 1.73e34·23-s − 1.74e35·24-s + 3.13e35·25-s − 7.88e35·26-s − 2.52e36·27-s + 1.67e37·28-s + ⋯
L(s)  = 1  − 1.83·2-s + 0.442·3-s + 2.37·4-s + 1.30·5-s − 0.812·6-s + 0.885·7-s − 2.51·8-s − 0.804·9-s − 2.39·10-s + 1.30·11-s + 1.04·12-s + 0.355·13-s − 1.62·14-s + 0.578·15-s + 2.24·16-s − 0.0816·17-s + 1.47·18-s − 0.710·19-s + 3.09·20-s + 0.392·21-s − 2.40·22-s + 0.328·23-s − 1.11·24-s + 0.705·25-s − 0.652·26-s − 0.798·27-s + 2.09·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(52-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+51/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(16.4731\)
Root analytic conductor: \(4.05871\)
Motivic weight: \(51\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :51/2),\ 1)\)

Particular Values

\(L(26)\) \(\approx\) \(1.324280379\)
\(L(\frac12)\) \(\approx\) \(1.324280379\)
\(L(\frac{53}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 8.71e7T + 2.25e15T^{2} \)
3 \( 1 - 6.49e11T + 2.15e24T^{2} \)
5 \( 1 - 8.70e17T + 4.44e35T^{2} \)
7 \( 1 - 3.14e21T + 1.25e43T^{2} \)
11 \( 1 - 4.69e26T + 1.29e53T^{2} \)
13 \( 1 - 9.04e27T + 6.47e56T^{2} \)
17 \( 1 + 1.94e30T + 5.66e62T^{2} \)
19 \( 1 + 2.88e32T + 1.64e65T^{2} \)
23 \( 1 - 1.73e34T + 2.80e69T^{2} \)
29 \( 1 - 2.51e37T + 3.82e74T^{2} \)
31 \( 1 + 3.39e37T + 1.14e76T^{2} \)
37 \( 1 - 1.41e40T + 9.51e79T^{2} \)
41 \( 1 - 1.62e41T + 1.78e82T^{2} \)
43 \( 1 + 1.06e41T + 2.02e83T^{2} \)
47 \( 1 + 2.33e42T + 1.89e85T^{2} \)
53 \( 1 + 5.52e43T + 8.67e87T^{2} \)
59 \( 1 - 2.66e45T + 2.05e90T^{2} \)
61 \( 1 + 3.00e45T + 1.12e91T^{2} \)
67 \( 1 - 1.24e45T + 1.34e93T^{2} \)
71 \( 1 - 2.67e47T + 2.59e94T^{2} \)
73 \( 1 - 4.01e47T + 1.07e95T^{2} \)
79 \( 1 + 3.09e48T + 6.01e96T^{2} \)
83 \( 1 - 1.48e48T + 7.46e97T^{2} \)
89 \( 1 + 1.49e49T + 2.62e99T^{2} \)
97 \( 1 + 5.46e50T + 2.11e101T^{2} \)
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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

−19.71047514777347790943703505461, −17.87033522073818177222113370622, −16.98003218471215958545650931905, −14.44902257838287693713433768040, −11.20306248452225429016226034819, −9.483691877095220456525194371351, −8.387233229994476295859626624605, −6.30812417909032547209546068873, −2.31336045986708990103866891156, −1.16938621946237143718274013481, 1.16938621946237143718274013481, 2.31336045986708990103866891156, 6.30812417909032547209546068873, 8.387233229994476295859626624605, 9.483691877095220456525194371351, 11.20306248452225429016226034819, 14.44902257838287693713433768040, 16.98003218471215958545650931905, 17.87033522073818177222113370622, 19.71047514777347790943703505461

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