↑

Properties

Label 2-2016-252.67-c0-0-1
Degree $2$
Conductor $2016$
Sign $-0.624 + 0.781i$
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 − i·7-s − 9-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s − 21-s − 2i·23-s − 25-s + i·27-s + (−0.5 − 0.866i)29-s + (−0.866 + 0.5i)31-s + (0.5 + 0.866i)37-s + (0.866 + 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  − i·3-s − i·7-s − 9-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s − 21-s − 2i·23-s − 25-s + i·27-s + (−0.5 − 0.866i)29-s + (−0.866 + 0.5i)31-s + (0.5 + 0.866i)37-s + (0.866 + 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.624 + 0.781i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ -0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9618742766\)
\(L(\frac12)\) \(\approx\) \(0.9618742766\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
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 \( 1 + 2iT - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.084233633422152801001414261535, −8.029887799836437857839354281332, −7.41770875037540658142914081284, −6.89061303902335712897684203167, −6.08862588194168505814805882055, −5.04124535716138313403161523463, −4.15661976152353882462500083534, −2.99007783825929830520150373683, −2.00519150011210948246172494671, −0.68237076840298874323420351541, 1.85783863167669148192655382561, 3.16086895813228209228601510753, 3.66572330014829490973476375430, 4.94186402873202804668246052107, 5.65067382945860378773458271663, 5.99481906651498941146789773439, 7.67238019784267588436961473066, 7.947531099478789812630669243034, 9.128582074111423844048283424491, 9.574826585056736883021466329284

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