Properties

Degree $2$
Conductor $8036$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.729·3-s − 3.02·5-s − 2.46·9-s − 2.63·11-s + 0.419·13-s − 2.20·15-s − 4.48·17-s + 4.82·19-s + 0.672·23-s + 4.13·25-s − 3.98·27-s − 7.40·29-s − 3.91·31-s − 1.92·33-s − 2.78·37-s + 0.306·39-s + 41-s + 9.85·43-s + 7.45·45-s − 12.2·47-s − 3.27·51-s − 12.6·53-s + 7.96·55-s + 3.52·57-s + 8.17·59-s − 5.69·61-s − 1.26·65-s + ⋯
L(s)  = 1  + 0.421·3-s − 1.35·5-s − 0.822·9-s − 0.794·11-s + 0.116·13-s − 0.569·15-s − 1.08·17-s + 1.10·19-s + 0.140·23-s + 0.826·25-s − 0.767·27-s − 1.37·29-s − 0.703·31-s − 0.334·33-s − 0.458·37-s + 0.0490·39-s + 0.156·41-s + 1.50·43-s + 1.11·45-s − 1.78·47-s − 0.458·51-s − 1.73·53-s + 1.07·55-s + 0.466·57-s + 1.06·59-s − 0.729·61-s − 0.157·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8036\)    =    \(2^{2} \cdot 7^{2} \cdot 41\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8036} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8036,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7170886853\)
\(L(\frac12)\) \(\approx\) \(0.7170886853\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 - 0.729T + 3T^{2} \)
5 \( 1 + 3.02T + 5T^{2} \)
11 \( 1 + 2.63T + 11T^{2} \)
13 \( 1 - 0.419T + 13T^{2} \)
17 \( 1 + 4.48T + 17T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 - 0.672T + 23T^{2} \)
29 \( 1 + 7.40T + 29T^{2} \)
31 \( 1 + 3.91T + 31T^{2} \)
37 \( 1 + 2.78T + 37T^{2} \)
43 \( 1 - 9.85T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 - 8.17T + 59T^{2} \)
61 \( 1 + 5.69T + 61T^{2} \)
67 \( 1 + 4.13T + 67T^{2} \)
71 \( 1 - 16.7T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 5.02T + 79T^{2} \)
83 \( 1 - 4.57T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 - 4.57T + 97T^{2} \)
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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

−7.969666985736344844494663399934, −7.34560496564638799848041231916, −6.61092626853661038009141890014, −5.60527563783350352395784062748, −5.04609553894315134455205788832, −4.12519500012417177425320274195, −3.47137261433355647716801072707, −2.86324929749222902594003691714, −1.89766137736111020700337289517, −0.38822452387893386965980257193, 0.38822452387893386965980257193, 1.89766137736111020700337289517, 2.86324929749222902594003691714, 3.47137261433355647716801072707, 4.12519500012417177425320274195, 5.04609553894315134455205788832, 5.60527563783350352395784062748, 6.61092626853661038009141890014, 7.34560496564638799848041231916, 7.969666985736344844494663399934

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