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Properties

Label 2-2016-252.247-c0-0-1
Degree $2$
Conductor $2016$
Sign $0.520 - 0.853i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
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  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.499 + 0.866i)21-s + (−1.73 + i)23-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (−0.5 − 0.866i)29-s − i·31-s + (0.5 − 0.866i)37-s + (−0.866 + 0.499i)39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.499 + 0.866i)21-s + (−1.73 + i)23-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (−0.5 − 0.866i)29-s − i·31-s + (0.5 − 0.866i)37-s + (−0.866 + 0.499i)39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.520 - 0.853i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (1759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.520 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.644764116\)
\(L(\frac12)\) \(\approx\) \(1.644764116\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.434660322986583375402480473757, −8.664189380374373608973294514691, −7.996340901360237788504101570999, −7.46516678112148866545569711897, −6.21479776660880744238447783882, −5.40775532470821949967799243742, −4.27088330587698693734407519958, −3.97418173718215582855768224897, −2.39903397860615386404388525549, −1.94791006495430479341127322606, 1.18272266625324960999058156770, 2.29165466064677525566454066428, 3.24666672671110643123638858651, 4.23525311016364615547633648254, 5.09435900690020940393654880118, 6.17349071297863209104639211345, 7.11220348976688838508196586923, 7.77717287355886891133890095812, 8.252083480123772611975164274657, 9.056951365201290765929043383380

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