Properties

Label 2-4017-1.1-c1-0-74
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  − 2.10·2-s + 3-s + 2.41·4-s − 2.56·5-s − 2.10·6-s − 4.38·7-s − 0.862·8-s + 9-s + 5.37·10-s − 4.74·11-s + 2.41·12-s + 13-s + 9.21·14-s − 2.56·15-s − 3.01·16-s + 5.66·17-s − 2.10·18-s + 2.69·19-s − 6.17·20-s − 4.38·21-s + 9.96·22-s + 2.40·23-s − 0.862·24-s + 1.56·25-s − 2.10·26-s + 27-s − 10.5·28-s + ⋯
L(s)  = 1  − 1.48·2-s + 0.577·3-s + 1.20·4-s − 1.14·5-s − 0.857·6-s − 1.65·7-s − 0.304·8-s + 0.333·9-s + 1.70·10-s − 1.43·11-s + 0.695·12-s + 0.277·13-s + 2.46·14-s − 0.661·15-s − 0.752·16-s + 1.37·17-s − 0.495·18-s + 0.618·19-s − 1.38·20-s − 0.957·21-s + 2.12·22-s + 0.501·23-s − 0.175·24-s + 0.312·25-s − 0.411·26-s + 0.192·27-s − 1.99·28-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 \( 1 + 2.10T + 2T^{2} \)
5 \( 1 + 2.56T + 5T^{2} \)
7 \( 1 + 4.38T + 7T^{2} \)
11 \( 1 + 4.74T + 11T^{2} \)
17 \( 1 - 5.66T + 17T^{2} \)
19 \( 1 - 2.69T + 19T^{2} \)
23 \( 1 - 2.40T + 23T^{2} \)
29 \( 1 - 4.48T + 29T^{2} \)
31 \( 1 + 9.26T + 31T^{2} \)
37 \( 1 - 3.61T + 37T^{2} \)
41 \( 1 - 2.01T + 41T^{2} \)
43 \( 1 - 0.321T + 43T^{2} \)
47 \( 1 - 6.05T + 47T^{2} \)
53 \( 1 - 4.84T + 53T^{2} \)
59 \( 1 + 4.97T + 59T^{2} \)
61 \( 1 + 9.70T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 5.24T + 71T^{2} \)
73 \( 1 + 2.68T + 73T^{2} \)
79 \( 1 + 2.72T + 79T^{2} \)
83 \( 1 - 3.79T + 83T^{2} \)
89 \( 1 - 3.53T + 89T^{2} \)
97 \( 1 - 8.12T + 97T^{2} \)
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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

−8.016966986058216035959827100597, −7.56527980956031557898272896716, −7.17676449131487810345523764176, −6.15005385907356058126642317113, −5.14505862261247205999112854521, −3.88072274809835171105636555333, −3.21498331903324496894460271547, −2.50835939092684474157744049666, −0.954770687595189254061385045420, 0, 0.954770687595189254061385045420, 2.50835939092684474157744049666, 3.21498331903324496894460271547, 3.88072274809835171105636555333, 5.14505862261247205999112854521, 6.15005385907356058126642317113, 7.17676449131487810345523764176, 7.56527980956031557898272896716, 8.016966986058216035959827100597

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