↑

Properties

Label 2-325-65.19-c2-0-0
Degree $2$
Conductor $325$
Sign $-0.900 + 0.435i$
Analytic cond. $8.85560$
Root an. cond. $2.97583$
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  + (−0.401 + 0.107i)2-s + (3.61 + 2.08i)3-s + (−3.31 + 1.91i)4-s + (−1.67 − 0.448i)6-s + (−10.0 − 2.68i)7-s + (2.30 − 2.30i)8-s + (4.19 + 7.26i)9-s + (−4.73 + 1.26i)11-s − 15.9·12-s + (−10.5 − 7.53i)13-s + 4.31·14-s + (6.97 − 12.0i)16-s + (11.0 + 19.1i)17-s + (−2.46 − 2.46i)18-s + (−33.5 − 8.98i)19-s + ⋯
L(s)  = 1  + (−0.200 + 0.0537i)2-s + (1.20 + 0.695i)3-s + (−0.828 + 0.478i)4-s + (−0.279 − 0.0747i)6-s + (−1.43 − 0.384i)7-s + (0.287 − 0.287i)8-s + (0.466 + 0.807i)9-s + (−0.430 + 0.115i)11-s − 1.33·12-s + (−0.814 − 0.579i)13-s + 0.308·14-s + (0.436 − 0.755i)16-s + (0.650 + 1.12i)17-s + (−0.137 − 0.137i)18-s + (−1.76 − 0.472i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.900 + 0.435i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.900 + 0.435i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0564709 - 0.246481i\)
\(L(\frac12)\) \(\approx\) \(0.0564709 - 0.246481i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (10.5 + 7.53i)T \)
good2 \( 1 + (0.401 - 0.107i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (-3.61 - 2.08i)T + (4.5 + 7.79i)T^{2} \)
7 \( 1 + (10.0 + 2.68i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (4.73 - 1.26i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-11.0 - 19.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (33.5 + 8.98i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (15.9 - 27.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-13.5 + 23.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-9.08 + 9.08i)T - 961iT^{2} \)
37 \( 1 + (-3.85 - 14.3i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (3.69 + 13.7i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-2.96 - 5.13i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (33.5 - 33.5i)T - 2.20e3iT^{2} \)
53 \( 1 - 33.8iT - 2.80e3T^{2} \)
59 \( 1 + (9.82 - 36.6i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (9.55 + 16.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (43.6 - 11.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (63.4 + 17.0i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (55.4 - 55.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 71.0T + 6.24e3T^{2} \)
83 \( 1 + (-54.0 - 54.0i)T + 6.88e3iT^{2} \)
89 \( 1 + (-66.8 + 17.9i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (14.8 - 55.3i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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

−12.24683614693737862733313264684, −10.30884961345025656865758174766, −9.985661043390721185834557489652, −9.180560378961059563929597897123, −8.273056523472659756227992600608, −7.56374491478973832991602435595, −6.09756222900311207514832752234, −4.45589907186406494829180355956, −3.65871249385705185845487279645, −2.73870129196895506211061227007, 0.10433657451386251545885793750, 2.13828765811660096215322534115, 3.21022428315294879976109562234, 4.65182939161743326660502325197, 6.10571978689230571230062670321, 7.09080636463719329762891142546, 8.291477501096560002412944596387, 8.909055712239132757059685613709, 9.747869316416961929185336654306, 10.42442410823925726060571208937

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