Properties

Label 2-4034-1.1-c1-0-75
Degree $2$
Conductor $4034$
Sign $1$
Analytic cond. $32.2116$
Root an. cond. $5.67553$
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 + 2.29·3-s + 4-s + 1.83·5-s − 2.29·6-s + 3.96·7-s − 8-s + 2.27·9-s − 1.83·10-s − 6.34·11-s + 2.29·12-s + 0.666·13-s − 3.96·14-s + 4.21·15-s + 16-s − 2.51·17-s − 2.27·18-s + 6.49·19-s + 1.83·20-s + 9.10·21-s + 6.34·22-s + 3.92·23-s − 2.29·24-s − 1.62·25-s − 0.666·26-s − 1.65·27-s + 3.96·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.821·5-s − 0.937·6-s + 1.49·7-s − 0.353·8-s + 0.759·9-s − 0.580·10-s − 1.91·11-s + 0.663·12-s + 0.184·13-s − 1.05·14-s + 1.08·15-s + 0.250·16-s − 0.610·17-s − 0.537·18-s + 1.49·19-s + 0.410·20-s + 1.98·21-s + 1.35·22-s + 0.818·23-s − 0.468·24-s − 0.325·25-s − 0.130·26-s − 0.318·27-s + 0.748·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4034\)    =    \(2 \cdot 2017\)
Sign: $1$
Analytic conductor: \(32.2116\)
Root analytic conductor: \(5.67553\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4034,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.990382709\)
\(L(\frac12)\) \(\approx\) \(2.990382709\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
2017 \( 1 - T \)
good3 \( 1 - 2.29T + 3T^{2} \)
5 \( 1 - 1.83T + 5T^{2} \)
7 \( 1 - 3.96T + 7T^{2} \)
11 \( 1 + 6.34T + 11T^{2} \)
13 \( 1 - 0.666T + 13T^{2} \)
17 \( 1 + 2.51T + 17T^{2} \)
19 \( 1 - 6.49T + 19T^{2} \)
23 \( 1 - 3.92T + 23T^{2} \)
29 \( 1 - 6.27T + 29T^{2} \)
31 \( 1 - 2.49T + 31T^{2} \)
37 \( 1 + 1.13T + 37T^{2} \)
41 \( 1 - 5.43T + 41T^{2} \)
43 \( 1 + 0.0452T + 43T^{2} \)
47 \( 1 + 6.48T + 47T^{2} \)
53 \( 1 + 4.67T + 53T^{2} \)
59 \( 1 + 0.176T + 59T^{2} \)
61 \( 1 - 9.50T + 61T^{2} \)
67 \( 1 - 8.80T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 - 9.00T + 73T^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 - 9.19T + 89T^{2} \)
97 \( 1 - 8.83T + 97T^{2} \)
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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

−8.321336953033901111117776111016, −7.911853542211432659802496533717, −7.49079179989781008964206489345, −6.39156621602934091227922393341, −5.21136062601359029160505032652, −4.97141183731482002743152350027, −3.49927548556074361633677585372, −2.51427616214161910107809869397, −2.20158624419290736200874533559, −1.08633256473544823085833895259, 1.08633256473544823085833895259, 2.20158624419290736200874533559, 2.51427616214161910107809869397, 3.49927548556074361633677585372, 4.97141183731482002743152350027, 5.21136062601359029160505032652, 6.39156621602934091227922393341, 7.49079179989781008964206489345, 7.911853542211432659802496533717, 8.321336953033901111117776111016

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