Properties

Label 2-7942-1.1-c1-0-6
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 − 2.39·3-s + 4-s − 0.391·5-s + 2.39·6-s − 1.71·7-s − 8-s + 2.71·9-s + 0.391·10-s − 11-s − 2.39·12-s − 3.71·13-s + 1.71·14-s + 0.935·15-s + 16-s + 5.43·17-s − 2.71·18-s − 0.391·20-s + 4.11·21-s + 22-s − 3.43·23-s + 2.39·24-s − 4.84·25-s + 3.71·26-s + 0.672·27-s − 1.71·28-s + 3.82·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.38·3-s + 0.5·4-s − 0.175·5-s + 0.976·6-s − 0.649·7-s − 0.353·8-s + 0.906·9-s + 0.123·10-s − 0.301·11-s − 0.690·12-s − 1.03·13-s + 0.459·14-s + 0.241·15-s + 0.250·16-s + 1.31·17-s − 0.640·18-s − 0.0875·20-s + 0.896·21-s + 0.213·22-s − 0.716·23-s + 0.488·24-s − 0.969·25-s + 0.729·26-s + 0.129·27-s − 0.324·28-s + 0.710·29-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

\(L(1)\) \(\approx\) \(0.2904961125\)
\(L(\frac12)\) \(\approx\) \(0.2904961125\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.39T + 3T^{2} \)
5 \( 1 + 0.391T + 5T^{2} \)
7 \( 1 + 1.71T + 7T^{2} \)
13 \( 1 + 3.71T + 13T^{2} \)
17 \( 1 - 5.43T + 17T^{2} \)
23 \( 1 + 3.43T + 23T^{2} \)
29 \( 1 - 3.82T + 29T^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 - 4.65T + 37T^{2} \)
41 \( 1 + 1.06T + 41T^{2} \)
43 \( 1 + 3.73T + 43T^{2} \)
47 \( 1 + 8.22T + 47T^{2} \)
53 \( 1 - 1.21T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 1.71T + 67T^{2} \)
71 \( 1 + 1.04T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 - 8.51T + 83T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 - 4.65T + 97T^{2} \)
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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

−7.86506342548505630558599105043, −7.09252305861686730828329194085, −6.42390929806197203726979148015, −5.90300803475461354973839020729, −5.21446528244165584270630327272, −4.50763448085514694389951685713, −3.41849510289572149103247860793, −2.59874359347062607742585882298, −1.40826110029927433936355435417, −0.33407510291370773399809401488, 0.33407510291370773399809401488, 1.40826110029927433936355435417, 2.59874359347062607742585882298, 3.41849510289572149103247860793, 4.50763448085514694389951685713, 5.21446528244165584270630327272, 5.90300803475461354973839020729, 6.42390929806197203726979148015, 7.09252305861686730828329194085, 7.86506342548505630558599105043

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