Properties

Label 2-4017-1.1-c1-0-64
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  − 2.64·2-s − 3-s + 5.00·4-s + 2.13·5-s + 2.64·6-s + 3.46·7-s − 7.95·8-s + 9-s − 5.65·10-s + 3.03·11-s − 5.00·12-s − 13-s − 9.16·14-s − 2.13·15-s + 11.0·16-s + 5.83·17-s − 2.64·18-s − 4.49·19-s + 10.6·20-s − 3.46·21-s − 8.03·22-s + 3.00·23-s + 7.95·24-s − 0.440·25-s + 2.64·26-s − 27-s + 17.3·28-s + ⋯
L(s)  = 1  − 1.87·2-s − 0.577·3-s + 2.50·4-s + 0.954·5-s + 1.08·6-s + 1.30·7-s − 2.81·8-s + 0.333·9-s − 1.78·10-s + 0.914·11-s − 1.44·12-s − 0.277·13-s − 2.44·14-s − 0.551·15-s + 2.75·16-s + 1.41·17-s − 0.623·18-s − 1.03·19-s + 2.38·20-s − 0.755·21-s − 1.71·22-s + 0.626·23-s + 1.62·24-s − 0.0881·25-s + 0.519·26-s − 0.192·27-s + 3.27·28-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

\(L(1)\) \(\approx\) \(1.033880689\)
\(L(\frac12)\) \(\approx\) \(1.033880689\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T + 2T^{2} \)
5 \( 1 - 2.13T + 5T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 3.03T + 11T^{2} \)
17 \( 1 - 5.83T + 17T^{2} \)
19 \( 1 + 4.49T + 19T^{2} \)
23 \( 1 - 3.00T + 23T^{2} \)
29 \( 1 + 1.98T + 29T^{2} \)
31 \( 1 + 4.26T + 31T^{2} \)
37 \( 1 + 9.10T + 37T^{2} \)
41 \( 1 - 9.24T + 41T^{2} \)
43 \( 1 + 4.21T + 43T^{2} \)
47 \( 1 - 1.62T + 47T^{2} \)
53 \( 1 + 2.06T + 53T^{2} \)
59 \( 1 - 4.74T + 59T^{2} \)
61 \( 1 - 6.40T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 + 3.28T + 71T^{2} \)
73 \( 1 - 8.31T + 73T^{2} \)
79 \( 1 - 5.66T + 79T^{2} \)
83 \( 1 - 9.21T + 83T^{2} \)
89 \( 1 + 6.21T + 89T^{2} \)
97 \( 1 - 7.11T + 97T^{2} \)
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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

−8.548443897439564693486670260689, −7.83213822196688699735544969333, −7.17696540204814054024923471203, −6.46492110630786359925364616003, −5.71874922315939102717662250782, −5.02603358977435907786230868942, −3.66092150344664741724732173039, −2.19877914470557630416268070653, −1.67940415752439977886863462223, −0.847350878004292842360125230647, 0.847350878004292842360125230647, 1.67940415752439977886863462223, 2.19877914470557630416268070653, 3.66092150344664741724732173039, 5.02603358977435907786230868942, 5.71874922315939102717662250782, 6.46492110630786359925364616003, 7.17696540204814054024923471203, 7.83213822196688699735544969333, 8.548443897439564693486670260689

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