Properties

Label 2-4017-1.1-c1-0-191
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.78·2-s + 3-s + 1.17·4-s − 1.00·5-s + 1.78·6-s − 0.242·7-s − 1.47·8-s + 9-s − 1.78·10-s − 3.69·11-s + 1.17·12-s + 13-s − 0.432·14-s − 1.00·15-s − 4.97·16-s + 6.00·17-s + 1.78·18-s − 3.61·19-s − 1.17·20-s − 0.242·21-s − 6.57·22-s − 4.86·23-s − 1.47·24-s − 3.99·25-s + 1.78·26-s + 27-s − 0.284·28-s + ⋯
L(s)  = 1  + 1.25·2-s + 0.577·3-s + 0.585·4-s − 0.448·5-s + 0.727·6-s − 0.0917·7-s − 0.521·8-s + 0.333·9-s − 0.565·10-s − 1.11·11-s + 0.338·12-s + 0.277·13-s − 0.115·14-s − 0.259·15-s − 1.24·16-s + 1.45·17-s + 0.419·18-s − 0.829·19-s − 0.262·20-s − 0.0529·21-s − 1.40·22-s − 1.01·23-s − 0.301·24-s − 0.798·25-s + 0.349·26-s + 0.192·27-s − 0.0537·28-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: \(1\)
Selberg data: \((2,\ 4017,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.78T + 2T^{2} \)
5 \( 1 + 1.00T + 5T^{2} \)
7 \( 1 + 0.242T + 7T^{2} \)
11 \( 1 + 3.69T + 11T^{2} \)
17 \( 1 - 6.00T + 17T^{2} \)
19 \( 1 + 3.61T + 19T^{2} \)
23 \( 1 + 4.86T + 23T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 + 0.861T + 31T^{2} \)
37 \( 1 - 4.72T + 37T^{2} \)
41 \( 1 + 2.97T + 41T^{2} \)
43 \( 1 + 5.03T + 43T^{2} \)
47 \( 1 + 3.48T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + 7.81T + 59T^{2} \)
61 \( 1 - 5.91T + 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + 3.92T + 73T^{2} \)
79 \( 1 + 3.94T + 79T^{2} \)
83 \( 1 + 9.51T + 83T^{2} \)
89 \( 1 + 17.6T + 89T^{2} \)
97 \( 1 + 1.85T + 97T^{2} \)
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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

−8.078718312222048485440688636448, −7.39884341487441175500852683388, −6.32452492686097173991373549748, −5.78767035182908915859325661592, −4.89455535717770722089630110257, −4.26499193380851548353165004156, −3.40592855797887332120622746870, −2.92811283491981945918184534711, −1.82328954642813310906763673038, 0, 1.82328954642813310906763673038, 2.92811283491981945918184534711, 3.40592855797887332120622746870, 4.26499193380851548353165004156, 4.89455535717770722089630110257, 5.78767035182908915859325661592, 6.32452492686097173991373549748, 7.39884341487441175500852683388, 8.078718312222048485440688636448

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