Properties

Label 2-7942-1.1-c1-0-206
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 + 1.56·3-s + 4-s + 2·5-s + 1.56·6-s + 3.56·7-s + 8-s − 0.561·9-s + 2·10-s + 11-s + 1.56·12-s + 3.56·13-s + 3.56·14-s + 3.12·15-s + 16-s + 3.56·17-s − 0.561·18-s + 2·20-s + 5.56·21-s + 22-s + 5.56·23-s + 1.56·24-s − 25-s + 3.56·26-s − 5.56·27-s + 3.56·28-s − 6.68·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.901·3-s + 0.5·4-s + 0.894·5-s + 0.637·6-s + 1.34·7-s + 0.353·8-s − 0.187·9-s + 0.632·10-s + 0.301·11-s + 0.450·12-s + 0.987·13-s + 0.951·14-s + 0.806·15-s + 0.250·16-s + 0.863·17-s − 0.132·18-s + 0.447·20-s + 1.21·21-s + 0.213·22-s + 1.15·23-s + 0.318·24-s − 0.200·25-s + 0.698·26-s − 1.07·27-s + 0.673·28-s − 1.24·29-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

\(L(1)\) \(\approx\) \(7.040029248\)
\(L(\frac12)\) \(\approx\) \(7.040029248\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
7 \( 1 - 3.56T + 7T^{2} \)
13 \( 1 - 3.56T + 13T^{2} \)
17 \( 1 - 3.56T + 17T^{2} \)
23 \( 1 - 5.56T + 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 3.12T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 7.80T + 53T^{2} \)
59 \( 1 + 4.68T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + 4.68T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 2.68T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 2.24T + 83T^{2} \)
89 \( 1 - 9.12T + 89T^{2} \)
97 \( 1 + 1.12T + 97T^{2} \)
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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

−7.76149139709409670564370199030, −7.30739420095091067637585896249, −6.26152210085320616205618021448, −5.58312156423369932347845145021, −5.20909408360791196359647595809, −4.16835199699085956052137885388, −3.51966767263986573898780279740, −2.72051812345261325163503029859, −1.84362520720394966377157797982, −1.32574849154556456165089571751, 1.32574849154556456165089571751, 1.84362520720394966377157797982, 2.72051812345261325163503029859, 3.51966767263986573898780279740, 4.16835199699085956052137885388, 5.20909408360791196359647595809, 5.58312156423369932347845145021, 6.26152210085320616205618021448, 7.30739420095091067637585896249, 7.76149139709409670564370199030

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