Properties

Label 2-4034-1.1-c1-0-137
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.523·3-s + 4-s + 1.37·5-s − 0.523·6-s − 3.92·7-s + 8-s − 2.72·9-s + 1.37·10-s + 1.18·11-s − 0.523·12-s + 5.13·13-s − 3.92·14-s − 0.720·15-s + 16-s − 3.19·17-s − 2.72·18-s − 6.02·19-s + 1.37·20-s + 2.05·21-s + 1.18·22-s + 8.56·23-s − 0.523·24-s − 3.10·25-s + 5.13·26-s + 2.99·27-s − 3.92·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.302·3-s + 0.5·4-s + 0.615·5-s − 0.213·6-s − 1.48·7-s + 0.353·8-s − 0.908·9-s + 0.435·10-s + 0.356·11-s − 0.151·12-s + 1.42·13-s − 1.04·14-s − 0.186·15-s + 0.250·16-s − 0.775·17-s − 0.642·18-s − 1.38·19-s + 0.307·20-s + 0.448·21-s + 0.251·22-s + 1.78·23-s − 0.106·24-s − 0.621·25-s + 1.00·26-s + 0.577·27-s − 0.741·28-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: \(1\)
Selberg data: \((2,\ 4034,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
2017 \( 1 - T \)
good3 \( 1 + 0.523T + 3T^{2} \)
5 \( 1 - 1.37T + 5T^{2} \)
7 \( 1 + 3.92T + 7T^{2} \)
11 \( 1 - 1.18T + 11T^{2} \)
13 \( 1 - 5.13T + 13T^{2} \)
17 \( 1 + 3.19T + 17T^{2} \)
19 \( 1 + 6.02T + 19T^{2} \)
23 \( 1 - 8.56T + 23T^{2} \)
29 \( 1 + 4.87T + 29T^{2} \)
31 \( 1 - 5.99T + 31T^{2} \)
37 \( 1 + 5.74T + 37T^{2} \)
41 \( 1 + 0.0599T + 41T^{2} \)
43 \( 1 + 9.15T + 43T^{2} \)
47 \( 1 - 5.61T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + 4.93T + 59T^{2} \)
61 \( 1 + 7.66T + 61T^{2} \)
67 \( 1 + 6.06T + 67T^{2} \)
71 \( 1 + 14.2T + 71T^{2} \)
73 \( 1 - 5.27T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 6.40T + 83T^{2} \)
89 \( 1 - 6.10T + 89T^{2} \)
97 \( 1 - 7.36T + 97T^{2} \)
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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

−8.141932470440259757955416571662, −6.81715933618855654125137248506, −6.41270617630214290827575547771, −6.04423513861634288015853888670, −5.21599345069883503458597926622, −4.20734869098385417997886196912, −3.35564433722888934555804290895, −2.75545202963653204719999362642, −1.56994652115675373683589604220, 0, 1.56994652115675373683589604220, 2.75545202963653204719999362642, 3.35564433722888934555804290895, 4.20734869098385417997886196912, 5.21599345069883503458597926622, 6.04423513861634288015853888670, 6.41270617630214290827575547771, 6.81715933618855654125137248506, 8.141932470440259757955416571662

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