Properties

Label 2-7942-1.1-c1-0-53
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 − 0.772·3-s + 4-s + 1.22·5-s + 0.772·6-s + 3.40·7-s − 8-s − 2.40·9-s − 1.22·10-s − 11-s − 0.772·12-s + 1.40·13-s − 3.40·14-s − 0.948·15-s + 16-s − 4.80·17-s + 2.40·18-s + 1.22·20-s − 2.62·21-s + 22-s + 6.80·23-s + 0.772·24-s − 3.49·25-s − 1.40·26-s + 4.17·27-s + 3.40·28-s − 8.03·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.446·3-s + 0.5·4-s + 0.548·5-s + 0.315·6-s + 1.28·7-s − 0.353·8-s − 0.800·9-s − 0.388·10-s − 0.301·11-s − 0.223·12-s + 0.389·13-s − 0.909·14-s − 0.244·15-s + 0.250·16-s − 1.16·17-s + 0.566·18-s + 0.274·20-s − 0.573·21-s + 0.213·22-s + 1.41·23-s + 0.157·24-s − 0.698·25-s − 0.275·26-s + 0.803·27-s + 0.643·28-s − 1.49·29-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

\(L(1)\) \(\approx\) \(1.293284835\)
\(L(\frac12)\) \(\approx\) \(1.293284835\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.772T + 3T^{2} \)
5 \( 1 - 1.22T + 5T^{2} \)
7 \( 1 - 3.40T + 7T^{2} \)
13 \( 1 - 1.40T + 13T^{2} \)
17 \( 1 + 4.80T + 17T^{2} \)
23 \( 1 - 6.80T + 23T^{2} \)
29 \( 1 + 8.03T + 29T^{2} \)
31 \( 1 - 4.94T + 31T^{2} \)
37 \( 1 + 2.35T + 37T^{2} \)
41 \( 1 + 2.94T + 41T^{2} \)
43 \( 1 + 9.12T + 43T^{2} \)
47 \( 1 - 5.25T + 47T^{2} \)
53 \( 1 - 4.45T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 3.40T + 67T^{2} \)
71 \( 1 - 7.57T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 - 3.71T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 + 14.9T + 89T^{2} \)
97 \( 1 + 2.35T + 97T^{2} \)
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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

−7.978854741165381665281653437908, −7.15906749726195594851163191986, −6.54665106083361868285376464920, −5.68669838526825236550310536567, −5.25250027381262449544238711551, −4.49174878561605892290820500977, −3.36751253372305631771039999052, −2.33269587707357491858031002617, −1.75155092248507693176569239097, −0.64118461406034651318597407850, 0.64118461406034651318597407850, 1.75155092248507693176569239097, 2.33269587707357491858031002617, 3.36751253372305631771039999052, 4.49174878561605892290820500977, 5.25250027381262449544238711551, 5.68669838526825236550310536567, 6.54665106083361868285376464920, 7.15906749726195594851163191986, 7.978854741165381665281653437908

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