↑

Properties

Label 2-912-19.12-c2-0-1
Degree $2$
Conductor $912$
Sign $-0.269 - 0.963i$
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 + (−1.64 − 2.85i)5-s − 7.47·7-s + (1.5 − 2.59i)9-s − 6.84·11-s + (13.8 + 7.98i)13-s + (−4.94 − 2.85i)15-s + (1.88 + 3.26i)17-s + (−18.7 − 2.89i)19-s + (−11.2 + 6.46i)21-s + (−6.40 + 11.0i)23-s + (7.06 − 12.2i)25-s − 5.19i·27-s + (−12.0 − 6.96i)29-s + 26.1i·31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.329 − 0.570i)5-s − 1.06·7-s + (0.166 − 0.288i)9-s − 0.622·11-s + (1.06 + 0.613i)13-s + (−0.329 − 0.190i)15-s + (0.110 + 0.191i)17-s + (−0.988 − 0.152i)19-s + (−0.533 + 0.308i)21-s + (−0.278 + 0.481i)23-s + (0.282 − 0.489i)25-s − 0.192i·27-s + (−0.415 − 0.240i)29-s + 0.843i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.269 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.269 - 0.963i$
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: \((2,\ 912,\ (\ :1),\ -0.269 - 0.963i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6641472259\)
\(L(\frac12)\) \(\approx\) \(0.6641472259\)
\(L(2)\) 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 \( 1 + (18.7 + 2.89i)T \)
good5 \( 1 + (1.64 + 2.85i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + 7.47T + 49T^{2} \)
11 \( 1 + 6.84T + 121T^{2} \)
13 \( 1 + (-13.8 - 7.98i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-1.88 - 3.26i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (6.40 - 11.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (12.0 + 6.96i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 26.1iT - 961T^{2} \)
37 \( 1 - 29.7iT - 1.36e3T^{2} \)
41 \( 1 + (58.6 - 33.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-25.5 - 44.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (9.60 - 16.6i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-72.6 - 41.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (59.0 - 34.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (40.2 - 69.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (24.6 + 14.2i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-11.9 + 6.92i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-9.46 - 16.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-129. + 74.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 26.7T + 6.88e3T^{2} \)
89 \( 1 + (-16.5 - 9.53i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (0.570 - 0.329i)T + (4.70e3 - 8.14e3i)T^{2} \)
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

−10.06495335958554558534101590804, −9.122476456953187692831962290802, −8.540505395953695803752912065498, −7.75375893656939196758331782122, −6.64426298453178027724172277442, −6.05307042301230861744223509410, −4.68672932995405765706143593267, −3.74329913320732618199027857084, −2.77913570719365854084948921917, −1.36026106859774920600763643257, 0.20143984355012814347976161692, 2.23661248063287928305382005675, 3.33495742509785072035233159553, 3.86371227862779877929656911381, 5.30953402111095752475602148644, 6.27789317986510950711627972999, 7.10512269699270149806450092749, 8.047508173609720552354986031037, 8.799365017092283013886868854871, 9.683590820546206721473018102983

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