Properties

Label 2-4017-1.1-c1-0-162
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  + 2.42·2-s + 3-s + 3.85·4-s + 0.700·5-s + 2.42·6-s + 1.99·7-s + 4.49·8-s + 9-s + 1.69·10-s + 4.15·11-s + 3.85·12-s + 13-s + 4.82·14-s + 0.700·15-s + 3.16·16-s + 2.15·17-s + 2.42·18-s − 7.77·19-s + 2.70·20-s + 1.99·21-s + 10.0·22-s + 3.42·23-s + 4.49·24-s − 4.50·25-s + 2.42·26-s + 27-s + 7.68·28-s + ⋯
L(s)  = 1  + 1.71·2-s + 0.577·3-s + 1.92·4-s + 0.313·5-s + 0.988·6-s + 0.752·7-s + 1.58·8-s + 0.333·9-s + 0.536·10-s + 1.25·11-s + 1.11·12-s + 0.277·13-s + 1.28·14-s + 0.180·15-s + 0.790·16-s + 0.522·17-s + 0.570·18-s − 1.78·19-s + 0.604·20-s + 0.434·21-s + 2.14·22-s + 0.713·23-s + 0.917·24-s − 0.901·25-s + 0.474·26-s + 0.192·27-s + 1.45·28-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

\(L(1)\) \(\approx\) \(8.223716420\)
\(L(\frac12)\) \(\approx\) \(8.223716420\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.42T + 2T^{2} \)
5 \( 1 - 0.700T + 5T^{2} \)
7 \( 1 - 1.99T + 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
17 \( 1 - 2.15T + 17T^{2} \)
19 \( 1 + 7.77T + 19T^{2} \)
23 \( 1 - 3.42T + 23T^{2} \)
29 \( 1 - 4.90T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 - 9.44T + 37T^{2} \)
41 \( 1 + 3.86T + 41T^{2} \)
43 \( 1 + 2.76T + 43T^{2} \)
47 \( 1 + 1.31T + 47T^{2} \)
53 \( 1 - 5.21T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 4.13T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 + 2.55T + 79T^{2} \)
83 \( 1 + 7.88T + 83T^{2} \)
89 \( 1 - 0.180T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−8.412476213851597810962786399693, −7.47848254057736792204395057027, −6.70690930767084105044640369566, −6.12433252947602520519952581675, −5.36980792871520412997053412426, −4.44747859021561073311984492679, −4.01986614993316586965393199726, −3.21198163195893959748572995774, −2.18516396078040097505578595640, −1.51384966508370624657672538836, 1.51384966508370624657672538836, 2.18516396078040097505578595640, 3.21198163195893959748572995774, 4.01986614993316586965393199726, 4.44747859021561073311984492679, 5.36980792871520412997053412426, 6.12433252947602520519952581675, 6.70690930767084105044640369566, 7.47848254057736792204395057027, 8.412476213851597810962786399693

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