↑

Properties

Label 2-912-19.12-c2-0-7
Degree $2$
Conductor $912$
Sign $-0.677 - 0.735i$
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 + (3.03 + 5.25i)5-s + 2.33·7-s + (1.5 − 2.59i)9-s − 12.7·11-s + (7.42 + 4.28i)13-s + (−9.10 − 5.25i)15-s + (11.9 + 20.7i)17-s + (18.0 − 5.96i)19-s + (−3.49 + 2.02i)21-s + (−8.34 + 14.4i)23-s + (−5.91 + 10.2i)25-s + 5.19i·27-s + (−15.3 − 8.86i)29-s + 29.5i·31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.606 + 1.05i)5-s + 0.333·7-s + (0.166 − 0.288i)9-s − 1.15·11-s + (0.570 + 0.329i)13-s + (−0.606 − 0.350i)15-s + (0.705 + 1.22i)17-s + (0.949 − 0.313i)19-s + (−0.166 + 0.0962i)21-s + (−0.362 + 0.628i)23-s + (−0.236 + 0.409i)25-s + 0.192i·27-s + (−0.529 − 0.305i)29-s + 0.952i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.677 - 0.735i$
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: \((2,\ 912,\ (\ :1),\ -0.677 - 0.735i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.420956932\)
\(L(\frac12)\) \(\approx\) \(1.420956932\)
\(L(2)\) 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 \( 1 + (-18.0 + 5.96i)T \)
good5 \( 1 + (-3.03 - 5.25i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 - 2.33T + 49T^{2} \)
11 \( 1 + 12.7T + 121T^{2} \)
13 \( 1 + (-7.42 - 4.28i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-11.9 - 20.7i)T + (-144.5 + 250. i)T^{2} \)
23 \( 1 + (8.34 - 14.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (15.3 + 8.86i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 29.5iT - 961T^{2} \)
37 \( 1 + 25.8iT - 1.36e3T^{2} \)
41 \( 1 + (28.6 - 16.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-30.3 - 52.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-1.46 + 2.54i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (40.1 + 23.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-93.0 + 53.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-21.4 + 37.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.21 + 1.85i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (95.4 - 55.1i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-53.3 - 92.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (105. - 60.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 112.T + 6.88e3T^{2} \)
89 \( 1 + (119. + 68.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-29.4 + 16.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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

−10.13023586080328895658281188855, −9.789462086695920094892113776022, −8.457619380589806733649838354361, −7.61696855804451922279703947218, −6.67526922276778119378629081948, −5.81833461930177543399161777170, −5.17606201391976506888712732956, −3.82192244519972237664777908020, −2.83123377441316455458644403897, −1.52533237150880618000408715830, 0.50261317715467303617706866611, 1.60830141525183993058712742366, 2.95098096261169322514206454310, 4.49941794230092651872693552617, 5.47424745993092248652777631337, 5.65346832311183680526781618774, 7.13673437073895107569970995714, 7.901683992064347989926151236099, 8.705379275925602207654374347661, 9.680133503105663695057024042165

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