Properties

Label 2-7942-1.1-c1-0-123
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 − 3.30·3-s + 4-s − 0.0759·5-s − 3.30·6-s − 4.78·7-s + 8-s + 7.92·9-s − 0.0759·10-s + 11-s − 3.30·12-s − 2.77·13-s − 4.78·14-s + 0.251·15-s + 16-s − 0.186·17-s + 7.92·18-s − 0.0759·20-s + 15.8·21-s + 22-s − 4.77·23-s − 3.30·24-s − 4.99·25-s − 2.77·26-s − 16.2·27-s − 4.78·28-s + 10.7·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.90·3-s + 0.5·4-s − 0.0339·5-s − 1.34·6-s − 1.80·7-s + 0.353·8-s + 2.64·9-s − 0.0240·10-s + 0.301·11-s − 0.954·12-s − 0.770·13-s − 1.27·14-s + 0.0648·15-s + 0.250·16-s − 0.0452·17-s + 1.86·18-s − 0.0169·20-s + 3.45·21-s + 0.213·22-s − 0.995·23-s − 0.674·24-s − 0.998·25-s − 0.544·26-s − 3.13·27-s − 0.904·28-s + 1.99·29-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 \( 1 + 3.30T + 3T^{2} \)
5 \( 1 + 0.0759T + 5T^{2} \)
7 \( 1 + 4.78T + 7T^{2} \)
13 \( 1 + 2.77T + 13T^{2} \)
17 \( 1 + 0.186T + 17T^{2} \)
23 \( 1 + 4.77T + 23T^{2} \)
29 \( 1 - 10.7T + 29T^{2} \)
31 \( 1 - 4.05T + 31T^{2} \)
37 \( 1 + 1.01T + 37T^{2} \)
41 \( 1 - 9.90T + 41T^{2} \)
43 \( 1 - 6.50T + 43T^{2} \)
47 \( 1 + 2.51T + 47T^{2} \)
53 \( 1 - 4.25T + 53T^{2} \)
59 \( 1 - 0.727T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 8.33T + 67T^{2} \)
71 \( 1 + 6.61T + 71T^{2} \)
73 \( 1 + 1.97T + 73T^{2} \)
79 \( 1 - 7.64T + 79T^{2} \)
83 \( 1 + 3.73T + 83T^{2} \)
89 \( 1 + 7.68T + 89T^{2} \)
97 \( 1 - 3.47T + 97T^{2} \)
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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

−7.12275341824116325969238916385, −6.47247909081030896547276796820, −6.04927455358057894590186398009, −5.69371803885386056785120214653, −4.56481510961077395356540282753, −4.28717681025604589402718189270, −3.28830281304737859597550950031, −2.31718505712939734505951837204, −0.945135623970580511951953551366, 0, 0.945135623970580511951953551366, 2.31718505712939734505951837204, 3.28830281304737859597550950031, 4.28717681025604589402718189270, 4.56481510961077395356540282753, 5.69371803885386056785120214653, 6.04927455358057894590186398009, 6.47247909081030896547276796820, 7.12275341824116325969238916385

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