Properties

Label 2-1-1.1-c47-0-0
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  − 2.02e7·2-s − 2.01e10·3-s + 2.71e14·4-s − 3.11e16·5-s + 4.09e17·6-s − 1.26e20·7-s − 2.64e21·8-s − 2.61e22·9-s + 6.32e23·10-s − 1.78e23·11-s − 5.46e24·12-s − 1.12e26·13-s + 2.56e27·14-s + 6.28e26·15-s + 1.55e28·16-s + 4.31e28·17-s + 5.31e29·18-s − 7.19e29·19-s − 8.45e30·20-s + 2.55e30·21-s + 3.62e30·22-s + 8.88e30·23-s + 5.34e31·24-s + 2.60e32·25-s + 2.28e33·26-s + 1.06e33·27-s − 3.43e34·28-s + ⋯
L(s)  = 1  − 1.71·2-s − 0.123·3-s + 1.92·4-s − 1.16·5-s + 0.211·6-s − 1.74·7-s − 1.58·8-s − 0.984·9-s + 2.00·10-s − 0.0601·11-s − 0.238·12-s − 0.746·13-s + 2.99·14-s + 0.144·15-s + 0.787·16-s + 0.523·17-s + 1.68·18-s − 0.640·19-s − 2.25·20-s + 0.216·21-s + 0.102·22-s + 0.0886·23-s + 0.196·24-s + 0.367·25-s + 1.27·26-s + 0.245·27-s − 3.36·28-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

\(L(24)\) \(\approx\) \(0.1274431610\)
\(L(\frac12)\) \(\approx\) \(0.1274431610\)
\(L(\frac{49}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 2.02e7T + 1.40e14T^{2} \)
3 \( 1 + 2.01e10T + 2.65e22T^{2} \)
5 \( 1 + 3.11e16T + 7.10e32T^{2} \)
7 \( 1 + 1.26e20T + 5.24e39T^{2} \)
11 \( 1 + 1.78e23T + 8.81e48T^{2} \)
13 \( 1 + 1.12e26T + 2.26e52T^{2} \)
17 \( 1 - 4.31e28T + 6.77e57T^{2} \)
19 \( 1 + 7.19e29T + 1.26e60T^{2} \)
23 \( 1 - 8.88e30T + 1.00e64T^{2} \)
29 \( 1 - 2.84e34T + 5.40e68T^{2} \)
31 \( 1 - 1.11e35T + 1.24e70T^{2} \)
37 \( 1 + 9.87e36T + 5.07e73T^{2} \)
41 \( 1 + 3.56e37T + 6.32e75T^{2} \)
43 \( 1 + 3.37e38T + 5.92e76T^{2} \)
47 \( 1 + 1.22e39T + 3.87e78T^{2} \)
53 \( 1 + 1.08e39T + 1.09e81T^{2} \)
59 \( 1 + 1.45e40T + 1.69e83T^{2} \)
61 \( 1 - 4.79e41T + 8.13e83T^{2} \)
67 \( 1 - 3.35e42T + 6.69e85T^{2} \)
71 \( 1 - 9.20e42T + 1.02e87T^{2} \)
73 \( 1 + 5.22e43T + 3.76e87T^{2} \)
79 \( 1 + 7.68e43T + 1.54e89T^{2} \)
83 \( 1 + 1.71e45T + 1.57e90T^{2} \)
89 \( 1 + 9.70e45T + 4.18e91T^{2} \)
97 \( 1 - 5.35e46T + 2.38e93T^{2} \)
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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

−19.78880197808427816036450246665, −19.15513542619149578736597245464, −16.96309481192772037816863607953, −15.73424110202675186102707082064, −11.94815917211133557240383180920, −10.07922282019894674193145975946, −8.415174341120272693876340904649, −6.75012880687945371681079481690, −2.99795514297973204710885199718, −0.32132544407402366206956330567, 0.32132544407402366206956330567, 2.99795514297973204710885199718, 6.75012880687945371681079481690, 8.415174341120272693876340904649, 10.07922282019894674193145975946, 11.94815917211133557240383180920, 15.73424110202675186102707082064, 16.96309481192772037816863607953, 19.15513542619149578736597245464, 19.78880197808427816036450246665

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