Properties

Label 2-4017-1.1-c1-0-113
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  + 2.38·2-s + 3-s + 3.67·4-s − 0.828·5-s + 2.38·6-s − 1.40·7-s + 3.98·8-s + 9-s − 1.97·10-s − 0.0373·11-s + 3.67·12-s − 13-s − 3.35·14-s − 0.828·15-s + 2.14·16-s + 5.38·17-s + 2.38·18-s + 6.09·19-s − 3.04·20-s − 1.40·21-s − 0.0889·22-s + 1.29·23-s + 3.98·24-s − 4.31·25-s − 2.38·26-s + 27-s − 5.17·28-s + ⋯
L(s)  = 1  + 1.68·2-s + 0.577·3-s + 1.83·4-s − 0.370·5-s + 0.972·6-s − 0.532·7-s + 1.40·8-s + 0.333·9-s − 0.624·10-s − 0.0112·11-s + 1.06·12-s − 0.277·13-s − 0.896·14-s − 0.213·15-s + 0.537·16-s + 1.30·17-s + 0.561·18-s + 1.39·19-s − 0.680·20-s − 0.307·21-s − 0.0189·22-s + 0.269·23-s + 0.813·24-s − 0.862·25-s − 0.467·26-s + 0.192·27-s − 0.977·28-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

\(L(1)\) \(\approx\) \(6.250564613\)
\(L(\frac12)\) \(\approx\) \(6.250564613\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.38T + 2T^{2} \)
5 \( 1 + 0.828T + 5T^{2} \)
7 \( 1 + 1.40T + 7T^{2} \)
11 \( 1 + 0.0373T + 11T^{2} \)
17 \( 1 - 5.38T + 17T^{2} \)
19 \( 1 - 6.09T + 19T^{2} \)
23 \( 1 - 1.29T + 23T^{2} \)
29 \( 1 - 8.82T + 29T^{2} \)
31 \( 1 - 0.864T + 31T^{2} \)
37 \( 1 - 2.68T + 37T^{2} \)
41 \( 1 - 8.48T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 7.40T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 2.27T + 59T^{2} \)
61 \( 1 - 7.94T + 61T^{2} \)
67 \( 1 + 3.11T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 + 3.48T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−8.192291577784498706456163839807, −7.41121788044683121146544351314, −7.00686019096151442368105011348, −5.84322959169985254688712641812, −5.56347541381709940771727827012, −4.41466372174522159171355748349, −3.96578024972535745937849244014, −2.92878665512132172824656034678, −2.74812319790157001277317418693, −1.16073749847165398389035504419, 1.16073749847165398389035504419, 2.74812319790157001277317418693, 2.92878665512132172824656034678, 3.96578024972535745937849244014, 4.41466372174522159171355748349, 5.56347541381709940771727827012, 5.84322959169985254688712641812, 7.00686019096151442368105011348, 7.41121788044683121146544351314, 8.192291577784498706456163839807

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