↑

Properties

Label 2-2016-504.349-c0-0-3
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + (−0.866 − 1.5i)5-s + (−0.5 + 0.866i)7-s − 9-s + (−1.5 + 0.866i)15-s − 1.73·19-s + (0.866 + 0.5i)21-s + (−0.5 − 0.866i)23-s + (−1 + 1.73i)25-s + i·27-s + 1.73·35-s + (0.866 + 1.5i)45-s + (−0.499 − 0.866i)49-s + 1.73i·57-s + (−0.866 + 1.5i)61-s + ⋯
L(s)  = 1  − i·3-s + (−0.866 − 1.5i)5-s + (−0.5 + 0.866i)7-s − 9-s + (−1.5 + 0.866i)15-s − 1.73·19-s + (0.866 + 0.5i)21-s + (−0.5 − 0.866i)23-s + (−1 + 1.73i)25-s + i·27-s + 1.73·35-s + (0.866 + 1.5i)45-s + (−0.499 − 0.866i)49-s + 1.73i·57-s + (−0.866 + 1.5i)61-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: $\chi_{2016} (1105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2823350417\)
\(L(\frac12)\) \(\approx\) \(0.2823350417\)
\(L(1)\) 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 \( 1 + iT \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.73T + T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.647411206980138381465705165731, −8.297300004459757088701577904519, −7.43712956030455217015995959460, −6.42235014588170698553902225224, −5.79396015450652300920784762577, −4.82678475130247601012579241675, −3.99459154307184575507785216814, −2.71223058310025654341473284945, −1.65955840105362133994615913747, −0.19243348647517592314199942792, 2.42722081617323713014139120395, 3.45416307822295363717741884160, 3.86033833943861476576256222963, 4.69887583654662759293457986078, 6.08078006835454921744205224245, 6.61091990761913899397965391914, 7.51217539614055953825721092661, 8.152383579183601744895738411521, 9.171191501496340590646438022849, 10.07691468342238087420183564003

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