Properties

Label 2-7942-1.1-c1-0-113
Degree $2$
Conductor $7942$
Sign $-1$
Analytic cond. $63.4171$
Root an. cond. $7.96349$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.405·3-s + 4-s − 2.02·5-s − 0.405·6-s − 4.23·7-s − 8-s − 2.83·9-s + 2.02·10-s + 11-s + 0.405·12-s + 3.21·13-s + 4.23·14-s − 0.819·15-s + 16-s − 2.00·17-s + 2.83·18-s − 2.02·20-s − 1.71·21-s − 22-s − 2.37·23-s − 0.405·24-s − 0.906·25-s − 3.21·26-s − 2.36·27-s − 4.23·28-s + 2.15·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.233·3-s + 0.5·4-s − 0.904·5-s − 0.165·6-s − 1.60·7-s − 0.353·8-s − 0.945·9-s + 0.639·10-s + 0.301·11-s + 0.116·12-s + 0.892·13-s + 1.13·14-s − 0.211·15-s + 0.250·16-s − 0.485·17-s + 0.668·18-s − 0.452·20-s − 0.374·21-s − 0.213·22-s − 0.494·23-s − 0.0827·24-s − 0.181·25-s − 0.631·26-s − 0.455·27-s − 0.800·28-s + 0.400·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7942\)    =    \(2 \cdot 11 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(63.4171\)
Root analytic conductor: \(7.96349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7942,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
11 \( 1 - T \)
19 \( 1 \)
good3 \( 1 - 0.405T + 3T^{2} \)
5 \( 1 + 2.02T + 5T^{2} \)
7 \( 1 + 4.23T + 7T^{2} \)
13 \( 1 - 3.21T + 13T^{2} \)
17 \( 1 + 2.00T + 17T^{2} \)
23 \( 1 + 2.37T + 23T^{2} \)
29 \( 1 - 2.15T + 29T^{2} \)
31 \( 1 - 6.20T + 31T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 - 3.97T + 41T^{2} \)
43 \( 1 - 5.64T + 43T^{2} \)
47 \( 1 + 2.21T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 1.13T + 59T^{2} \)
61 \( 1 - 2.99T + 61T^{2} \)
67 \( 1 - 1.86T + 67T^{2} \)
71 \( 1 - 4.23T + 71T^{2} \)
73 \( 1 + 5.29T + 73T^{2} \)
79 \( 1 - 5.14T + 79T^{2} \)
83 \( 1 + 6.45T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + 18.9T + 97T^{2} \)
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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

−7.77742661410757894873894057024, −6.73614418346268713427285976603, −6.27486302094056769900781606887, −5.79101111831183977628952393649, −4.41052891024045094835887814889, −3.71978049235899281861621199381, −3.05711745461597912381811617581, −2.40457716172714322768270611456, −0.902421650037717782136201899671, 0, 0.902421650037717782136201899671, 2.40457716172714322768270611456, 3.05711745461597912381811617581, 3.71978049235899281861621199381, 4.41052891024045094835887814889, 5.79101111831183977628952393649, 6.27486302094056769900781606887, 6.73614418346268713427285976603, 7.77742661410757894873894057024

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