Properties

Label 2-4017-1.1-c1-0-57
Degree $2$
Conductor $4017$
Sign $1$
Analytic cond. $32.0759$
Root an. cond. $5.66355$
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  + 1.68·2-s − 3-s + 0.838·4-s + 0.142·5-s − 1.68·6-s − 0.0282·7-s − 1.95·8-s + 9-s + 0.239·10-s + 0.586·11-s − 0.838·12-s − 13-s − 0.0475·14-s − 0.142·15-s − 4.97·16-s + 6.55·17-s + 1.68·18-s + 3.66·19-s + 0.119·20-s + 0.0282·21-s + 0.988·22-s − 4.69·23-s + 1.95·24-s − 4.97·25-s − 1.68·26-s − 27-s − 0.0236·28-s + ⋯
L(s)  = 1  + 1.19·2-s − 0.577·3-s + 0.419·4-s + 0.0635·5-s − 0.687·6-s − 0.0106·7-s − 0.692·8-s + 0.333·9-s + 0.0756·10-s + 0.176·11-s − 0.241·12-s − 0.277·13-s − 0.0127·14-s − 0.0366·15-s − 1.24·16-s + 1.58·17-s + 0.397·18-s + 0.839·19-s + 0.0266·20-s + 0.00616·21-s + 0.210·22-s − 0.978·23-s + 0.399·24-s − 0.995·25-s − 0.330·26-s − 0.192·27-s − 0.00447·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $1$
Analytic conductor: \(32.0759\)
Root analytic conductor: \(5.66355\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4017,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.613950835\)
\(L(\frac12)\) \(\approx\) \(2.613950835\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 + T \)
103 \( 1 - T \)
good2 \( 1 - 1.68T + 2T^{2} \)
5 \( 1 - 0.142T + 5T^{2} \)
7 \( 1 + 0.0282T + 7T^{2} \)
11 \( 1 - 0.586T + 11T^{2} \)
17 \( 1 - 6.55T + 17T^{2} \)
19 \( 1 - 3.66T + 19T^{2} \)
23 \( 1 + 4.69T + 23T^{2} \)
29 \( 1 - 0.00650T + 29T^{2} \)
31 \( 1 - 7.78T + 31T^{2} \)
37 \( 1 + 0.640T + 37T^{2} \)
41 \( 1 + 0.302T + 41T^{2} \)
43 \( 1 - 1.22T + 43T^{2} \)
47 \( 1 - 8.93T + 47T^{2} \)
53 \( 1 + 2.65T + 53T^{2} \)
59 \( 1 - 1.29T + 59T^{2} \)
61 \( 1 - 14.7T + 61T^{2} \)
67 \( 1 + 7.28T + 67T^{2} \)
71 \( 1 - 2.54T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 - 4.48T + 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−8.223211451267048168411648881030, −7.62666210521742465969465357918, −6.65522966683615635284546179136, −5.97447508774920514632288074991, −5.43526385225421081811583030484, −4.78566745291081711971807903052, −3.90082897999479801045639501129, −3.29310620749203784334154137119, −2.19833661768419961218594240069, −0.800322514097457673707539341003, 0.800322514097457673707539341003, 2.19833661768419961218594240069, 3.29310620749203784334154137119, 3.90082897999479801045639501129, 4.78566745291081711971807903052, 5.43526385225421081811583030484, 5.97447508774920514632288074991, 6.65522966683615635284546179136, 7.62666210521742465969465357918, 8.223211451267048168411648881030

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