Properties

Label 2-4034-1.1-c1-0-52
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 − 2.36·3-s + 4-s − 2.34·5-s − 2.36·6-s + 4.14·7-s + 8-s + 2.61·9-s − 2.34·10-s + 0.755·11-s − 2.36·12-s + 5.20·13-s + 4.14·14-s + 5.54·15-s + 16-s − 0.343·17-s + 2.61·18-s + 3.14·19-s − 2.34·20-s − 9.80·21-s + 0.755·22-s + 3.00·23-s − 2.36·24-s + 0.480·25-s + 5.20·26-s + 0.915·27-s + 4.14·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.36·3-s + 0.5·4-s − 1.04·5-s − 0.967·6-s + 1.56·7-s + 0.353·8-s + 0.871·9-s − 0.740·10-s + 0.227·11-s − 0.683·12-s + 1.44·13-s + 1.10·14-s + 1.43·15-s + 0.250·16-s − 0.0833·17-s + 0.616·18-s + 0.721·19-s − 0.523·20-s − 2.14·21-s + 0.160·22-s + 0.625·23-s − 0.483·24-s + 0.0961·25-s + 1.02·26-s + 0.176·27-s + 0.782·28-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

\(L(1)\) \(\approx\) \(2.034228800\)
\(L(\frac12)\) \(\approx\) \(2.034228800\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.36T + 3T^{2} \)
5 \( 1 + 2.34T + 5T^{2} \)
7 \( 1 - 4.14T + 7T^{2} \)
11 \( 1 - 0.755T + 11T^{2} \)
13 \( 1 - 5.20T + 13T^{2} \)
17 \( 1 + 0.343T + 17T^{2} \)
19 \( 1 - 3.14T + 19T^{2} \)
23 \( 1 - 3.00T + 23T^{2} \)
29 \( 1 - 0.0845T + 29T^{2} \)
31 \( 1 + 1.30T + 31T^{2} \)
37 \( 1 + 3.42T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 1.80T + 43T^{2} \)
47 \( 1 - 7.76T + 47T^{2} \)
53 \( 1 + 3.31T + 53T^{2} \)
59 \( 1 + 6.69T + 59T^{2} \)
61 \( 1 - 6.64T + 61T^{2} \)
67 \( 1 - 8.31T + 67T^{2} \)
71 \( 1 + 6.11T + 71T^{2} \)
73 \( 1 + 0.836T + 73T^{2} \)
79 \( 1 + 5.99T + 79T^{2} \)
83 \( 1 - 4.33T + 83T^{2} \)
89 \( 1 - 8.23T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−8.280891474085046200393277625049, −7.55555666893812845266205060689, −6.87586627665854227994969448346, −6.04126705662491812035446684872, −5.33523478290438634946449619401, −4.79875137975934154802335008305, −4.07289328036816783561467266160, −3.31766830049038337792593564294, −1.72308328740186053985930678356, −0.845868362028569138764444024614, 0.845868362028569138764444024614, 1.72308328740186053985930678356, 3.31766830049038337792593564294, 4.07289328036816783561467266160, 4.79875137975934154802335008305, 5.33523478290438634946449619401, 6.04126705662491812035446684872, 6.87586627665854227994969448346, 7.55555666893812845266205060689, 8.280891474085046200393277625049

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