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Properties

Label 2-3017-3017.1056-c0-0-0
Degree $2$
Conductor $3017$
Sign $0.0906 + 0.995i$
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  + (1.70 − 0.514i)2-s + (1.82 − 1.20i)4-s + (−0.841 − 0.539i)7-s + (1.35 − 1.62i)8-s + (0.252 − 0.967i)9-s + (−0.0124 + 0.00770i)11-s + (−1.71 − 0.489i)14-s + (0.624 − 1.46i)16-s + (−0.0652 − 1.78i)18-s + (−0.0172 + 0.0195i)22-s + (−0.461 − 0.208i)23-s + (0.972 + 0.231i)25-s + (−2.18 + 0.0319i)28-s + (−1.09 + 0.0478i)29-s + (0.0800 − 0.727i)32-s + ⋯
L(s)  = 1  + (1.70 − 0.514i)2-s + (1.82 − 1.20i)4-s + (−0.841 − 0.539i)7-s + (1.35 − 1.62i)8-s + (0.252 − 0.967i)9-s + (−0.0124 + 0.00770i)11-s + (−1.71 − 0.489i)14-s + (0.624 − 1.46i)16-s + (−0.0652 − 1.78i)18-s + (−0.0172 + 0.0195i)22-s + (−0.461 − 0.208i)23-s + (0.972 + 0.231i)25-s + (−2.18 + 0.0319i)28-s + (−1.09 + 0.0478i)29-s + (0.0800 − 0.727i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0906 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0906 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3017\)    =    \(7 \cdot 431\)
Sign: $0.0906 + 0.995i$
Analytic conductor: \(1.50567\)
Root analytic conductor: \(1.22706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3017} (1056, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3017,\ (\ :0),\ 0.0906 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.127096178\)
\(L(\frac12)\) \(\approx\) \(3.127096178\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.841 + 0.539i)T \)
431 \( 1 + (0.976 + 0.217i)T \)
good2 \( 1 + (-1.70 + 0.514i)T + (0.833 - 0.551i)T^{2} \)
3 \( 1 + (-0.252 + 0.967i)T^{2} \)
5 \( 1 + (-0.972 - 0.231i)T^{2} \)
11 \( 1 + (0.0124 - 0.00770i)T + (0.444 - 0.895i)T^{2} \)
13 \( 1 + (0.267 - 0.963i)T^{2} \)
17 \( 1 + (0.714 + 0.699i)T^{2} \)
19 \( 1 + (0.923 + 0.384i)T^{2} \)
23 \( 1 + (0.461 + 0.208i)T + (0.661 + 0.749i)T^{2} \)
29 \( 1 + (1.09 - 0.0478i)T + (0.996 - 0.0875i)T^{2} \)
31 \( 1 + (0.295 + 0.955i)T^{2} \)
37 \( 1 + (0.0570 - 0.0848i)T + (-0.377 - 0.925i)T^{2} \)
41 \( 1 + (-0.984 - 0.174i)T^{2} \)
43 \( 1 + (-1.94 - 0.345i)T + (0.939 + 0.343i)T^{2} \)
47 \( 1 + (-0.639 - 0.768i)T^{2} \)
53 \( 1 + (-0.212 - 1.70i)T + (-0.969 + 0.245i)T^{2} \)
59 \( 1 + (0.350 - 0.936i)T^{2} \)
61 \( 1 + (0.672 + 0.739i)T^{2} \)
67 \( 1 + (0.294 + 0.375i)T + (-0.238 + 0.971i)T^{2} \)
71 \( 1 + (0.325 + 0.589i)T + (-0.533 + 0.845i)T^{2} \)
73 \( 1 + (0.994 - 0.102i)T^{2} \)
79 \( 1 + (-0.156 - 0.327i)T + (-0.628 + 0.777i)T^{2} \)
83 \( 1 + (0.841 + 0.539i)T^{2} \)
89 \( 1 + (-0.948 - 0.315i)T^{2} \)
97 \( 1 + (-0.0511 - 0.998i)T^{2} \)
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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

−8.999341927886884672228210486119, −7.56744417670441841611847780929, −6.91325549120720237327148117499, −6.20679836596572909424010564894, −5.68896112986711653766572983396, −4.58846981671712739451655308292, −3.96337256903113708118478648235, −3.31895413749694745161010098961, −2.52105519654414068909800526682, −1.17613464717580345760107885428, 2.08011089975294479392426672505, 2.81990981547643144445761364618, 3.72527238812889289994795685921, 4.46655451517107592385129376402, 5.37642516821474277142144303378, 5.77978047454438652953526715814, 6.67750164696217579450642318139, 7.26073713315021230218148524870, 8.014618894285410952321884411556, 8.956540633892811150038277010577

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