Properties

Label 2-7942-1.1-c1-0-126
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 2.16·3-s + 4-s + 4.16·5-s − 2.16·6-s − 0.683·7-s − 8-s + 1.68·9-s − 4.16·10-s − 11-s + 2.16·12-s − 2.68·13-s + 0.683·14-s + 9.01·15-s + 16-s + 3.36·17-s − 1.68·18-s + 4.16·20-s − 1.48·21-s + 22-s − 1.36·23-s − 2.16·24-s + 12.3·25-s + 2.68·26-s − 2.84·27-s − 0.683·28-s − 2.79·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.24·3-s + 0.5·4-s + 1.86·5-s − 0.883·6-s − 0.258·7-s − 0.353·8-s + 0.561·9-s − 1.31·10-s − 0.301·11-s + 0.624·12-s − 0.744·13-s + 0.182·14-s + 2.32·15-s + 0.250·16-s + 0.816·17-s − 0.396·18-s + 0.931·20-s − 0.323·21-s + 0.213·22-s − 0.285·23-s − 0.441·24-s + 2.46·25-s + 0.526·26-s − 0.548·27-s − 0.129·28-s − 0.519·29-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: $\chi_{7942} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7942,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.330308103\)
\(L(\frac12)\) \(\approx\) \(3.330308103\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.16T + 3T^{2} \)
5 \( 1 - 4.16T + 5T^{2} \)
7 \( 1 + 0.683T + 7T^{2} \)
13 \( 1 + 2.68T + 13T^{2} \)
17 \( 1 - 3.36T + 17T^{2} \)
23 \( 1 + 1.36T + 23T^{2} \)
29 \( 1 + 2.79T + 29T^{2} \)
31 \( 1 + 5.01T + 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 - 7.01T + 41T^{2} \)
43 \( 1 - 7.86T + 43T^{2} \)
47 \( 1 - 2.96T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 9.92T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 0.683T + 67T^{2} \)
71 \( 1 + 3.53T + 71T^{2} \)
73 \( 1 - 0.407T + 73T^{2} \)
79 \( 1 - 7.28T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−7.80452579579060155060282644353, −7.45352653711558250578624010935, −6.52571432279369548904526881768, −5.77109712612254622740352961694, −5.36085089733899590334804972438, −4.11711298645048061117821976669, −3.07269041421224312029660906665, −2.42547993265815295911299553356, −2.06195749525812658896100631395, −0.955507814206067275387571053669, 0.955507814206067275387571053669, 2.06195749525812658896100631395, 2.42547993265815295911299553356, 3.07269041421224312029660906665, 4.11711298645048061117821976669, 5.36085089733899590334804972438, 5.77109712612254622740352961694, 6.52571432279369548904526881768, 7.45352653711558250578624010935, 7.80452579579060155060282644353

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