↑

Properties

Label 2-495-55.2-c1-0-2
Degree $2$
Conductor $495$
Sign $0.231 - 0.972i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
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  + (−0.665 − 1.30i)2-s + (−0.0875 + 0.120i)4-s + (1.11 + 1.93i)5-s + (−4.16 + 0.659i)7-s + (−2.68 − 0.424i)8-s + (1.78 − 2.74i)10-s + (0.920 + 3.18i)11-s + (−0.824 + 0.420i)13-s + (3.63 + 4.99i)14-s + (1.32 + 4.06i)16-s + (−1.71 − 0.875i)17-s + (−4.39 + 3.19i)19-s + (−0.331 − 0.0347i)20-s + (3.54 − 3.32i)22-s + (−1.95 + 1.95i)23-s + ⋯
L(s)  = 1  + (−0.470 − 0.923i)2-s + (−0.0437 + 0.0602i)4-s + (0.500 + 0.865i)5-s + (−1.57 + 0.249i)7-s + (−0.947 − 0.150i)8-s + (0.564 − 0.869i)10-s + (0.277 + 0.960i)11-s + (−0.228 + 0.116i)13-s + (0.970 + 1.33i)14-s + (0.330 + 1.01i)16-s + (−0.416 − 0.212i)17-s + (−1.00 + 0.732i)19-s + (−0.0741 − 0.00776i)20-s + (0.756 − 0.708i)22-s + (−0.408 + 0.408i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.231 - 0.972i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.231 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.384932 + 0.303937i\)
\(L(\frac12)\) \(\approx\) \(0.384932 + 0.303937i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.11 - 1.93i)T \)
11 \( 1 + (-0.920 - 3.18i)T \)
good2 \( 1 + (0.665 + 1.30i)T + (-1.17 + 1.61i)T^{2} \)
7 \( 1 + (4.16 - 0.659i)T + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (0.824 - 0.420i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (1.71 + 0.875i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (4.39 - 3.19i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.95 - 1.95i)T - 23iT^{2} \)
29 \( 1 + (0.810 + 0.588i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.131 + 0.403i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.771 + 4.87i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (0.339 + 0.467i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.05 - 5.05i)T + 43iT^{2} \)
47 \( 1 + (-1.17 - 0.186i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-4.12 - 8.09i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (5.47 - 7.53i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-7.40 + 2.40i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.05 + 3.05i)T + 67iT^{2} \)
71 \( 1 + (2.65 + 8.17i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.854 + 5.39i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (0.705 - 2.17i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.902 - 1.77i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 + (2.96 - 1.50i)T + (57.0 - 78.4i)T^{2} \)
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

−10.85644726392440833728586904112, −10.21792475862988219419529528921, −9.583177439596917230999560717514, −9.041626484736504959830257022167, −7.35932005169061530885077515115, −6.43791845289729731287022965930, −5.88552130997724601416691818427, −3.95151603920713481396652976399, −2.84240882948133792157368236870, −1.98774119170315219977306418399, 0.31291262220054531562811908072, 2.66765905067187923321853071276, 3.95264506314316954968669197163, 5.49437320946730199342611305466, 6.36327691687863857007720427004, 6.85604750824920627496099915842, 8.234434193072979993297901996668, 8.857417044690429557845291725119, 9.539505793061348942031137612647, 10.47182492968122977069973840510

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