↑

Properties

Label 2-912-19.12-c2-0-12
Degree $2$
Conductor $912$
Sign $0.995 + 0.0902i$
Analytic cond. $24.8502$
Root an. cond. $4.98499$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)3-s + (0.268 + 0.464i)5-s − 12.8·7-s + (1.5 − 2.59i)9-s + 13.8·11-s + (10.2 + 5.91i)13-s + (0.804 + 0.464i)15-s + (−1.04 − 1.80i)17-s + (−9.37 + 16.5i)19-s + (−19.2 + 11.1i)21-s + (2.59 − 4.48i)23-s + (12.3 − 21.4i)25-s − 5.19i·27-s + (22.5 + 13.0i)29-s − 10.4i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.0536 + 0.0928i)5-s − 1.83·7-s + (0.166 − 0.288i)9-s + 1.25·11-s + (0.788 + 0.454i)13-s + (0.0536 + 0.0309i)15-s + (−0.0611 − 0.105i)17-s + (−0.493 + 0.869i)19-s + (−0.916 + 0.528i)21-s + (0.112 − 0.195i)23-s + (0.494 − 0.856i)25-s − 0.192i·27-s + (0.778 + 0.449i)29-s − 0.335i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.995 + 0.0902i$
Analytic conductor: \(24.8502\)
Root analytic conductor: \(4.98499\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1),\ 0.995 + 0.0902i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.009635221\)
\(L(\frac12)\) \(\approx\) \(2.009635221\)
\(L(2)\) 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 \( 1 + (-1.5 + 0.866i)T \)
19 \( 1 + (9.37 - 16.5i)T \)
good5 \( 1 + (-0.268 - 0.464i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 12.8T + 49T^{2} \)
11 \( 1 - 13.8T + 121T^{2} \)
13 \( 1 + (-10.2 - 5.91i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (1.04 + 1.80i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (-2.59 + 4.48i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-22.5 - 13.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 10.4iT - 961T^{2} \)
37 \( 1 - 70.6iT - 1.36e3T^{2} \)
41 \( 1 + (-69.8 + 40.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (25.7 + 44.5i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-20.3 + 35.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-80.9 - 46.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-39.5 + 22.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-21.4 + 37.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-106. - 61.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (63.7 - 36.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (10.5 + 18.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (7.69 - 4.44i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 23.8T + 6.88e3T^{2} \)
89 \( 1 + (106. + 61.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-54.7 + 31.6i)T + (4.70e3 - 8.14e3i)T^{2} \)
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

−9.867353487452132533592798745710, −8.934742761594921325580266219746, −8.515069949607163014113011932386, −7.01953806916390757692290912876, −6.58084724244818134031196664418, −5.90050947306907708724953698916, −4.13491359013061759793214411532, −3.52887725436903287533494106359, −2.43377791692440566872772069535, −0.919236684073863613223590765157, 0.868033298031013022573698434251, 2.62707369667509282140665609305, 3.51774459579181457749541185086, 4.25021759722788032258799835062, 5.74324337154015159805556258547, 6.50416244718694792105855749355, 7.20242238406221380018843188526, 8.518642725991104575811644349166, 9.216460661545295454218468930884, 9.638398220844560085706588171428

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