Properties

Label 2-4017-1.1-c1-0-17
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  − 0.365·2-s + 3-s − 1.86·4-s + 0.232·5-s − 0.365·6-s − 3.73·7-s + 1.41·8-s + 9-s − 0.0851·10-s − 5.65·11-s − 1.86·12-s − 13-s + 1.36·14-s + 0.232·15-s + 3.21·16-s + 4.10·17-s − 0.365·18-s − 2.34·19-s − 0.434·20-s − 3.73·21-s + 2.06·22-s − 0.356·23-s + 1.41·24-s − 4.94·25-s + 0.365·26-s + 27-s + 6.96·28-s + ⋯
L(s)  = 1  − 0.258·2-s + 0.577·3-s − 0.933·4-s + 0.104·5-s − 0.149·6-s − 1.41·7-s + 0.499·8-s + 0.333·9-s − 0.0269·10-s − 1.70·11-s − 0.538·12-s − 0.277·13-s + 0.364·14-s + 0.0601·15-s + 0.803·16-s + 0.996·17-s − 0.0861·18-s − 0.538·19-s − 0.0971·20-s − 0.814·21-s + 0.441·22-s − 0.0743·23-s + 0.288·24-s − 0.989·25-s + 0.0717·26-s + 0.192·27-s + 1.31·28-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

\(L(1)\) \(\approx\) \(0.7360842117\)
\(L(\frac12)\) \(\approx\) \(0.7360842117\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.365T + 2T^{2} \)
5 \( 1 - 0.232T + 5T^{2} \)
7 \( 1 + 3.73T + 7T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
17 \( 1 - 4.10T + 17T^{2} \)
19 \( 1 + 2.34T + 19T^{2} \)
23 \( 1 + 0.356T + 23T^{2} \)
29 \( 1 - 0.228T + 29T^{2} \)
31 \( 1 + 0.559T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 + 3.90T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 5.60T + 53T^{2} \)
59 \( 1 + 8.73T + 59T^{2} \)
61 \( 1 - 3.69T + 61T^{2} \)
67 \( 1 + 6.64T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 0.662T + 79T^{2} \)
83 \( 1 - 4.24T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 - 3.24T + 97T^{2} \)
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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

−8.560859243092841939591652176693, −7.67416438845734281272216354145, −7.36987157442841276639703856720, −6.11259886886871050508587734340, −5.49755227192909420818997731332, −4.65046456053079978656409728874, −3.65438450956458626652990327363, −3.07841361108544420472384053503, −2.08633809110084503018174836185, −0.47491461212212149117243697832, 0.47491461212212149117243697832, 2.08633809110084503018174836185, 3.07841361108544420472384053503, 3.65438450956458626652990327363, 4.65046456053079978656409728874, 5.49755227192909420818997731332, 6.11259886886871050508587734340, 7.36987157442841276639703856720, 7.67416438845734281272216354145, 8.560859243092841939591652176693

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