Properties

Label 2-83-83.82-c12-0-52
Degree $2$
Conductor $83$
Sign $1$
Analytic cond. $75.8614$
Root an. cond. $8.70984$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 835.·3-s + 4.09e3·4-s + 1.51e5·7-s + 1.66e5·9-s + 2.91e6·11-s − 3.42e6·12-s + 1.67e7·16-s + 4.76e7·17-s − 1.26e8·21-s − 2.31e8·23-s + 2.44e8·25-s + 3.04e8·27-s + 6.22e8·28-s + 7.35e7·29-s − 1.48e9·31-s − 2.43e9·33-s + 6.82e8·36-s + 1.55e9·37-s − 9.47e9·41-s + 1.19e10·44-s − 1.40e10·48-s + 9.22e9·49-s − 3.97e10·51-s + 8.40e10·59-s + 5.36e9·61-s + 2.53e10·63-s + 6.87e10·64-s + ⋯
L(s)  = 1  − 1.14·3-s + 4-s + 1.29·7-s + 0.313·9-s + 1.64·11-s − 1.14·12-s + 16-s + 1.97·17-s − 1.47·21-s − 1.56·23-s + 25-s + 0.786·27-s + 1.29·28-s + 0.123·29-s − 1.67·31-s − 1.88·33-s + 0.313·36-s + 0.605·37-s − 1.99·41-s + 1.64·44-s − 1.14·48-s + 0.666·49-s − 2.26·51-s + 1.99·59-s + 0.104·61-s + 0.404·63-s + 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(83\)
Sign: $1$
Analytic conductor: \(75.8614\)
Root analytic conductor: \(8.70984\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (82, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 83,\ (\ :6),\ 1)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.665566759\)
\(L(\frac12)\) \(\approx\) \(2.665566759\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 - 3.26e11T \)
good2 \( 1 - 4.09e3T^{2} \)
3 \( 1 + 835.T + 5.31e5T^{2} \)
5 \( 1 - 2.44e8T^{2} \)
7 \( 1 - 1.51e5T + 1.38e10T^{2} \)
11 \( 1 - 2.91e6T + 3.13e12T^{2} \)
13 \( 1 - 2.32e13T^{2} \)
17 \( 1 - 4.76e7T + 5.82e14T^{2} \)
19 \( 1 - 2.21e15T^{2} \)
23 \( 1 + 2.31e8T + 2.19e16T^{2} \)
29 \( 1 - 7.35e7T + 3.53e17T^{2} \)
31 \( 1 + 1.48e9T + 7.87e17T^{2} \)
37 \( 1 - 1.55e9T + 6.58e18T^{2} \)
41 \( 1 + 9.47e9T + 2.25e19T^{2} \)
43 \( 1 - 3.99e19T^{2} \)
47 \( 1 - 1.16e20T^{2} \)
53 \( 1 - 4.91e20T^{2} \)
59 \( 1 - 8.40e10T + 1.77e21T^{2} \)
61 \( 1 - 5.36e9T + 2.65e21T^{2} \)
67 \( 1 - 8.18e21T^{2} \)
71 \( 1 - 1.64e22T^{2} \)
73 \( 1 - 2.29e22T^{2} \)
79 \( 1 - 5.90e22T^{2} \)
89 \( 1 - 2.46e23T^{2} \)
97 \( 1 - 6.93e23T^{2} \)
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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

−11.81405687740170523787975394119, −11.09308532006320970152662817772, −10.04106536208030923888578737051, −8.326131608108892350463358363015, −7.14313871505069713934019061293, −6.06462331461534684840056413365, −5.18902716524000367416630308271, −3.63434061434975935541261509677, −1.76134289431743208580118491072, −0.973435885860746539521554841406, 0.973435885860746539521554841406, 1.76134289431743208580118491072, 3.63434061434975935541261509677, 5.18902716524000367416630308271, 6.06462331461534684840056413365, 7.14313871505069713934019061293, 8.326131608108892350463358363015, 10.04106536208030923888578737051, 11.09308532006320970152662817772, 11.81405687740170523787975394119

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