↑

Properties

Label 2-912-57.8-c1-0-24
Degree $2$
Conductor $912$
Sign $0.786 - 0.617i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.66 + 0.484i)3-s + (1.30 + 0.753i)5-s + 3.02·7-s + (2.53 + 1.61i)9-s + 4.73i·11-s + (2.73 − 1.57i)13-s + (1.80 + 1.88i)15-s + (−3.56 − 2.05i)17-s + (−3.27 − 2.87i)19-s + (5.03 + 1.46i)21-s + (−5.45 + 3.14i)23-s + (−1.36 − 2.36i)25-s + (3.42 + 3.90i)27-s + (−3.81 − 6.61i)29-s + 7.27i·31-s + ⋯
L(s)  = 1  + (0.960 + 0.279i)3-s + (0.583 + 0.337i)5-s + 1.14·7-s + (0.843 + 0.536i)9-s + 1.42i·11-s + (0.757 − 0.437i)13-s + (0.466 + 0.486i)15-s + (−0.864 − 0.499i)17-s + (−0.751 − 0.659i)19-s + (1.09 + 0.319i)21-s + (−1.13 + 0.656i)23-s + (−0.272 − 0.472i)25-s + (0.659 + 0.751i)27-s + (−0.708 − 1.22i)29-s + 1.30i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.786 - 0.617i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.786 - 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.54964 + 0.880355i\)
\(L(\frac12)\) \(\approx\) \(2.54964 + 0.880355i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 - 0.484i)T \)
19 \( 1 + (3.27 + 2.87i)T \)
good5 \( 1 + (-1.30 - 0.753i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.02T + 7T^{2} \)
11 \( 1 - 4.73iT - 11T^{2} \)
13 \( 1 + (-2.73 + 1.57i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.56 + 2.05i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.45 - 3.14i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.81 + 6.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.27iT - 31T^{2} \)
37 \( 1 - 1.63iT - 37T^{2} \)
41 \( 1 + (-5.98 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.45 + 9.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.43 - 1.40i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.08 - 3.61i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.79 - 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.68 - 6.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.81 + 1.62i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.03 + 1.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.24 - 2.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.72 + 3.30i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.7iT - 83T^{2} \)
89 \( 1 + (-1.41 - 2.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.24 + 3.60i)T + (48.5 + 84.0i)T^{2} \)
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

−10.21235542334433144466743273788, −9.261970560572611258039238812533, −8.602214560794201850521277612401, −7.68024531975197902137020986839, −7.03259046953252939666796491479, −5.78037439364793711941608452726, −4.63807386435891491943229746832, −3.99912073847092198683245031892, −2.40607422300525320737290048234, −1.85897152373873232979913421591, 1.38279100441750586941246119901, 2.22044537301976258109483935058, 3.65856295512578033393248546179, 4.47001191607127629663281059396, 5.83892527077973190402195685724, 6.44637262006062044335274307509, 7.894605705673500591050869559039, 8.299099196297874257478950888857, 8.967032797189072162878547525341, 9.796418340467969618408041413397

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