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Properties

Label 2-2016-504.13-c0-0-1
Degree $2$
Conductor $2016$
Sign $0.766 - 0.642i$
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  − 3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 9-s + (−1 + 1.73i)13-s + (−0.5 + 0.866i)15-s + 19-s + (−0.5 − 0.866i)21-s + (−0.5 + 0.866i)23-s − 27-s + 0.999·35-s + (1 − 1.73i)39-s + (0.5 − 0.866i)45-s + (−0.499 + 0.866i)49-s − 57-s + ⋯
L(s)  = 1  − 3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 9-s + (−1 + 1.73i)13-s + (−0.5 + 0.866i)15-s + 19-s + (−0.5 − 0.866i)21-s + (−0.5 + 0.866i)23-s − 27-s + 0.999·35-s + (1 − 1.73i)39-s + (0.5 − 0.866i)45-s + (−0.499 + 0.866i)49-s − 57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (1777, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9195158592\)
\(L(\frac12)\) \(\approx\) \(0.9195158592\)
\(L(1)\) 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 \( 1 + T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.502292939021935849968833983698, −8.873927735994536045742357422062, −7.79175624485030497880648951909, −6.98008603048691797400664182752, −6.12937839043021128594464501824, −5.18321900207771631888132484252, −5.00406147519657545248018539903, −3.92558591569212454969345732664, −2.21422115055156297333245059835, −1.41202221636522855859412981941, 0.823696630509385061811572228356, 2.32831327546849469838326384662, 3.41216833929191555446901271093, 4.57169095972395556697477761574, 5.31056646024587491272783934299, 6.03416293671113023274621689927, 6.96255467422517623922256287535, 7.46671474779637508401016785960, 8.229427129811690447597198501498, 9.681476190920630217860263815112

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