Properties

Label 2-4017-1.1-c1-0-87
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  + 1.15·2-s + 3-s − 0.655·4-s + 1.84·5-s + 1.15·6-s + 0.698·7-s − 3.07·8-s + 9-s + 2.13·10-s + 0.388·11-s − 0.655·12-s − 13-s + 0.809·14-s + 1.84·15-s − 2.25·16-s + 5.14·17-s + 1.15·18-s + 1.17·19-s − 1.20·20-s + 0.698·21-s + 0.450·22-s + 2.50·23-s − 3.07·24-s − 1.60·25-s − 1.15·26-s + 27-s − 0.458·28-s + ⋯
L(s)  = 1  + 0.819·2-s + 0.577·3-s − 0.327·4-s + 0.823·5-s + 0.473·6-s + 0.263·7-s − 1.08·8-s + 0.333·9-s + 0.675·10-s + 0.117·11-s − 0.189·12-s − 0.277·13-s + 0.216·14-s + 0.475·15-s − 0.564·16-s + 1.24·17-s + 0.273·18-s + 0.269·19-s − 0.270·20-s + 0.152·21-s + 0.0960·22-s + 0.522·23-s − 0.628·24-s − 0.321·25-s − 0.227·26-s + 0.192·27-s − 0.0865·28-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

\(L(1)\) \(\approx\) \(3.867510799\)
\(L(\frac12)\) \(\approx\) \(3.867510799\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.15T + 2T^{2} \)
5 \( 1 - 1.84T + 5T^{2} \)
7 \( 1 - 0.698T + 7T^{2} \)
11 \( 1 - 0.388T + 11T^{2} \)
17 \( 1 - 5.14T + 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 - 2.50T + 23T^{2} \)
29 \( 1 - 7.84T + 29T^{2} \)
31 \( 1 - 0.668T + 31T^{2} \)
37 \( 1 - 1.83T + 37T^{2} \)
41 \( 1 - 0.981T + 41T^{2} \)
43 \( 1 + 7.35T + 43T^{2} \)
47 \( 1 + 1.19T + 47T^{2} \)
53 \( 1 - 6.03T + 53T^{2} \)
59 \( 1 + 4.69T + 59T^{2} \)
61 \( 1 + 5.47T + 61T^{2} \)
67 \( 1 - 7.71T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 - 17.4T + 79T^{2} \)
83 \( 1 - 3.30T + 83T^{2} \)
89 \( 1 - 7.80T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−8.352987962746470796775641553869, −7.893309396253168981758843995920, −6.76438133486295951988175697024, −6.13266624329649731461614124131, −5.21744408862807015949148664230, −4.85158942664065178521567971477, −3.77046367806067099552652412776, −3.11209966007511107756815607353, −2.23059159410367274011863686363, −1.02358543540102969884618772498, 1.02358543540102969884618772498, 2.23059159410367274011863686363, 3.11209966007511107756815607353, 3.77046367806067099552652412776, 4.85158942664065178521567971477, 5.21744408862807015949148664230, 6.13266624329649731461614124131, 6.76438133486295951988175697024, 7.893309396253168981758843995920, 8.352987962746470796775641553869

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