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Properties

Label 2-3017-3017.1000-c0-0-0
Degree $2$
Conductor $3017$
Sign $0.647 - 0.762i$
Analytic cond. $1.50567$
Root an. cond. $1.22706$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.558 + 1.63i)2-s + (−1.57 − 1.21i)4-s + (−0.969 + 0.245i)7-s + (1.43 − 0.946i)8-s + (0.639 − 0.768i)9-s + (−1.93 + 0.313i)11-s + (0.138 − 1.72i)14-s + (0.246 + 0.943i)16-s + (0.901 + 1.47i)18-s + (0.566 − 3.33i)22-s + (0.977 + 0.825i)23-s + (0.965 − 0.259i)25-s + (1.82 + 0.792i)28-s + (−0.605 − 1.69i)29-s + (0.0290 + 0.00212i)32-s + ⋯
L(s)  = 1  + (−0.558 + 1.63i)2-s + (−1.57 − 1.21i)4-s + (−0.969 + 0.245i)7-s + (1.43 − 0.946i)8-s + (0.639 − 0.768i)9-s + (−1.93 + 0.313i)11-s + (0.138 − 1.72i)14-s + (0.246 + 0.943i)16-s + (0.901 + 1.47i)18-s + (0.566 − 3.33i)22-s + (0.977 + 0.825i)23-s + (0.965 − 0.259i)25-s + (1.82 + 0.792i)28-s + (−0.605 − 1.69i)29-s + (0.0290 + 0.00212i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3017\)    =    \(7 \cdot 431\)
Sign: $0.647 - 0.762i$
Analytic conductor: \(1.50567\)
Root analytic conductor: \(1.22706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3017} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3017,\ (\ :0),\ 0.647 - 0.762i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5574720211\)
\(L(\frac12)\) \(\approx\) \(0.5574720211\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.969 - 0.245i)T \)
431 \( 1 + (-0.989 - 0.145i)T \)
good2 \( 1 + (0.558 - 1.63i)T + (-0.791 - 0.611i)T^{2} \)
3 \( 1 + (-0.639 + 0.768i)T^{2} \)
5 \( 1 + (-0.965 + 0.259i)T^{2} \)
11 \( 1 + (1.93 - 0.313i)T + (0.948 - 0.315i)T^{2} \)
13 \( 1 + (-0.281 + 0.959i)T^{2} \)
17 \( 1 + (0.953 + 0.302i)T^{2} \)
19 \( 1 + (-0.0511 - 0.998i)T^{2} \)
23 \( 1 + (-0.977 - 0.825i)T + (0.167 + 0.985i)T^{2} \)
29 \( 1 + (0.605 + 1.69i)T + (-0.773 + 0.634i)T^{2} \)
31 \( 1 + (0.508 - 0.861i)T^{2} \)
37 \( 1 + (0.180 - 0.210i)T + (-0.152 - 0.988i)T^{2} \)
41 \( 1 + (-0.195 - 0.980i)T^{2} \)
43 \( 1 + (0.228 + 1.14i)T + (-0.923 + 0.384i)T^{2} \)
47 \( 1 + (-0.833 - 0.551i)T^{2} \)
53 \( 1 + (-1.58 - 0.555i)T + (0.782 + 0.622i)T^{2} \)
59 \( 1 + (0.825 - 0.563i)T^{2} \)
61 \( 1 + (0.238 - 0.971i)T^{2} \)
67 \( 1 + (-1.73 + 0.413i)T + (0.892 - 0.450i)T^{2} \)
71 \( 1 + (0.0430 + 1.96i)T + (-0.999 + 0.0438i)T^{2} \)
73 \( 1 + (0.961 - 0.274i)T^{2} \)
79 \( 1 + (-0.779 + 0.0684i)T + (0.984 - 0.174i)T^{2} \)
83 \( 1 + (0.969 - 0.245i)T^{2} \)
89 \( 1 + (0.911 + 0.411i)T^{2} \)
97 \( 1 + (-0.138 - 0.990i)T^{2} \)
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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

−8.916488925677358499603452962688, −8.094581657590988465888148052732, −7.37989227082866944026218067823, −6.92380916663307557462945792522, −6.12080283768080931859629328785, −5.44448829639987507053024915233, −4.76776501205909290568539922057, −3.58718773116630620645237298241, −2.46875780105561755202305360491, −0.52543928301438562554200381910, 0.982434090320232939170567771905, 2.33416903362895155668154128193, 2.89237697517106596981177162467, 3.67754412850144189177341848595, 4.78246215326178338334123650545, 5.38418577658846000589830613628, 6.80753833805066658815564494312, 7.47884985370781736043629591639, 8.406370759104904751033994460341, 8.912653872584269315315742181841

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