Properties

Label 2-4034-1.1-c1-0-18
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 − 2.58·3-s + 4-s − 1.78·5-s − 2.58·6-s − 0.188·7-s + 8-s + 3.69·9-s − 1.78·10-s − 4.86·11-s − 2.58·12-s + 5.13·13-s − 0.188·14-s + 4.61·15-s + 16-s − 4.65·17-s + 3.69·18-s − 2.98·19-s − 1.78·20-s + 0.488·21-s − 4.86·22-s − 1.55·23-s − 2.58·24-s − 1.81·25-s + 5.13·26-s − 1.80·27-s − 0.188·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.49·3-s + 0.5·4-s − 0.797·5-s − 1.05·6-s − 0.0712·7-s + 0.353·8-s + 1.23·9-s − 0.564·10-s − 1.46·11-s − 0.747·12-s + 1.42·13-s − 0.0504·14-s + 1.19·15-s + 0.250·16-s − 1.12·17-s + 0.871·18-s − 0.685·19-s − 0.398·20-s + 0.106·21-s − 1.03·22-s − 0.324·23-s − 0.528·24-s − 0.363·25-s + 1.00·26-s − 0.346·27-s − 0.0356·28-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

\(L(1)\) \(\approx\) \(0.8589191572\)
\(L(\frac12)\) \(\approx\) \(0.8589191572\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.58T + 3T^{2} \)
5 \( 1 + 1.78T + 5T^{2} \)
7 \( 1 + 0.188T + 7T^{2} \)
11 \( 1 + 4.86T + 11T^{2} \)
13 \( 1 - 5.13T + 13T^{2} \)
17 \( 1 + 4.65T + 17T^{2} \)
19 \( 1 + 2.98T + 19T^{2} \)
23 \( 1 + 1.55T + 23T^{2} \)
29 \( 1 + 4.32T + 29T^{2} \)
31 \( 1 - 0.532T + 31T^{2} \)
37 \( 1 - 9.81T + 37T^{2} \)
41 \( 1 - 9.33T + 41T^{2} \)
43 \( 1 + 0.538T + 43T^{2} \)
47 \( 1 + 9.10T + 47T^{2} \)
53 \( 1 + 4.35T + 53T^{2} \)
59 \( 1 + 9.08T + 59T^{2} \)
61 \( 1 + 3.15T + 61T^{2} \)
67 \( 1 - 6.33T + 67T^{2} \)
71 \( 1 - 3.02T + 71T^{2} \)
73 \( 1 + 0.972T + 73T^{2} \)
79 \( 1 - 7.94T + 79T^{2} \)
83 \( 1 - 3.66T + 83T^{2} \)
89 \( 1 - 1.04T + 89T^{2} \)
97 \( 1 - 9.63T + 97T^{2} \)
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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

−8.045612018136221290465343454969, −7.73358198844324153083530195045, −6.55348764930405062844751814814, −6.19995025578549975384719835624, −5.50858653376896883591133192460, −4.64940670586380149388651858454, −4.17819574776936598102109618643, −3.16864961889441305558980387766, −1.96861360164092975392168404700, −0.49520176791182296344021716408, 0.49520176791182296344021716408, 1.96861360164092975392168404700, 3.16864961889441305558980387766, 4.17819574776936598102109618643, 4.64940670586380149388651858454, 5.50858653376896883591133192460, 6.19995025578549975384719835624, 6.55348764930405062844751814814, 7.73358198844324153083530195045, 8.045612018136221290465343454969

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