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Properties

Label 2-2016-504.331-c0-0-0
Degree $2$
Conductor $2016$
Sign $0.592 - 0.805i$
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  + (−0.5 + 0.866i)3-s − i·5-s + (−0.866 + 0.5i)7-s + (−0.499 − 0.866i)9-s + 11-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + i·23-s + 0.999·27-s + (0.866 + 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s − i·5-s + (−0.866 + 0.5i)7-s + (−0.499 − 0.866i)9-s + 11-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + i·23-s + 0.999·27-s + (0.866 + 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.592 - 0.805i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.592 - 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8793641042\)
\(L(\frac12)\) \(\approx\) \(0.8793641042\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + iT - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.445945152751709103913269081609, −9.019468759000739541660634001980, −8.124544393240271169348170231442, −6.84955571761500196011968810678, −6.22460456980847296309167819874, −5.31698334931643136284447599685, −4.68454247411267010978399217822, −3.81699787375568658284012750972, −2.86593308409431831868101582571, −1.17547170775951359378833764723, 0.817736743427391074172127168236, 2.42166866857855211184263223157, 3.14855807483563938264413936434, 4.26687380725758997780366898412, 5.49908594279532170122419041114, 6.23860088721004461599308027952, 6.99582357201763928991412936509, 7.28765057660322562984065724973, 8.225704487792616832225642411587, 9.362489498476224087173471487711

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