Properties

Label 2-15e2-45.34-c3-0-2
Degree $2$
Conductor $225$
Sign $-0.863 - 0.504i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.195 − 0.112i)2-s + (4.76 + 2.06i)3-s + (−3.97 − 6.88i)4-s + (−0.697 − 0.940i)6-s + (−27.0 − 15.5i)7-s + 3.59i·8-s + (18.4 + 19.7i)9-s + (−9.06 + 15.6i)11-s + (−4.71 − 41.0i)12-s + (−43.4 + 25.0i)13-s + (3.51 + 6.08i)14-s + (−31.3 + 54.3i)16-s + 131. i·17-s + (−1.37 − 5.92i)18-s − 23.2·19-s + ⋯
L(s)  = 1  + (−0.0689 − 0.0398i)2-s + (0.917 + 0.397i)3-s + (−0.496 − 0.860i)4-s + (−0.0474 − 0.0639i)6-s + (−1.45 − 0.842i)7-s + 0.158i·8-s + (0.683 + 0.730i)9-s + (−0.248 + 0.430i)11-s + (−0.113 − 0.987i)12-s + (−0.926 + 0.535i)13-s + (0.0670 + 0.116i)14-s + (−0.490 + 0.849i)16-s + 1.87i·17-s + (−0.0180 − 0.0775i)18-s − 0.280·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.863 - 0.504i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.863 - 0.504i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0743464 + 0.274313i\)
\(L(\frac12)\) \(\approx\) \(0.0743464 + 0.274313i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.76 - 2.06i)T \)
5 \( 1 \)
good2 \( 1 + (0.195 + 0.112i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (27.0 + 15.5i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (9.06 - 15.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (43.4 - 25.0i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 131. iT - 4.91e3T^{2} \)
19 \( 1 + 23.2T + 6.85e3T^{2} \)
23 \( 1 + (28.5 - 16.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-62.9 + 108. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (62.5 + 108. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 99.9iT - 5.06e4T^{2} \)
41 \( 1 + (122. + 212. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (120. + 69.5i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (409. + 236. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 421. iT - 1.48e5T^{2} \)
59 \( 1 + (371. + 642. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (4.48 - 7.77i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (510. - 294. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 48.5T + 3.57e5T^{2} \)
73 \( 1 - 409. iT - 3.89e5T^{2} \)
79 \( 1 + (-265. + 459. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-255. - 147. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 852.T + 7.04e5T^{2} \)
97 \( 1 + (336. + 194. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−12.56018545602574601744434675544, −10.70937581704657263557948336907, −10.01105357530770131535180426935, −9.580734926560585612550440495827, −8.460517489634615855365017227051, −7.19816081819531535563174828171, −6.10199020427788550462580206990, −4.53464952758434273606333788611, −3.66096047227895136994257850128, −1.97695070491866759926222717975, 0.10101403321573117547451063540, 2.80475735010639126259529715744, 3.18049723084222477391136450905, 4.92429724126672196440205905578, 6.56384534852875528924785655976, 7.47001616256819371107745208568, 8.509975962879386877958240317270, 9.319644300060022137067750558330, 9.929026475909170029422777103815, 11.85533733232965599094429268003

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