↑

Properties

Label 2-15e2-9.4-c3-0-34
Degree $2$
Conductor $225$
Sign $0.820 + 0.571i$
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  + (2.42 + 4.20i)2-s + (−4.79 + 1.99i)3-s + (−7.76 + 13.4i)4-s + (−20.0 − 15.3i)6-s + (−8.11 − 14.0i)7-s − 36.5·8-s + (19.0 − 19.1i)9-s + (−29.7 − 51.5i)11-s + (10.4 − 80.0i)12-s + (−13.5 + 23.4i)13-s + (39.3 − 68.2i)14-s + (−26.5 − 45.9i)16-s + 78.9·17-s + (126. + 33.5i)18-s − 142.·19-s + ⋯
L(s)  = 1  + (0.857 + 1.48i)2-s + (−0.923 + 0.384i)3-s + (−0.970 + 1.68i)4-s + (−1.36 − 1.04i)6-s + (−0.438 − 0.759i)7-s − 1.61·8-s + (0.704 − 0.709i)9-s + (−0.815 − 1.41i)11-s + (0.250 − 1.92i)12-s + (−0.288 + 0.500i)13-s + (0.751 − 1.30i)14-s + (−0.414 − 0.718i)16-s + 1.12·17-s + (1.65 + 0.438i)18-s − 1.71·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.820 + 0.571i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.383346 - 0.120216i\)
\(L(\frac12)\) \(\approx\) \(0.383346 - 0.120216i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.79 - 1.99i)T \)
5 \( 1 \)
good2 \( 1 + (-2.42 - 4.20i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (8.11 + 14.0i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (29.7 + 51.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (13.5 - 23.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 78.9T + 4.91e3T^{2} \)
19 \( 1 + 142.T + 6.85e3T^{2} \)
23 \( 1 + (42.8 - 74.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (113. + 195. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-75.3 + 130. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 234.T + 5.06e4T^{2} \)
41 \( 1 + (36.9 - 63.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (133. + 231. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (133. + 230. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 603.T + 1.48e5T^{2} \)
59 \( 1 + (254. - 440. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (39.1 + 67.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (244. - 422. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 73.2T + 3.57e5T^{2} \)
73 \( 1 - 115.T + 3.89e5T^{2} \)
79 \( 1 + (391. + 677. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-135. - 235. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 211.T + 7.04e5T^{2} \)
97 \( 1 + (833. + 1.44e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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

−11.88606468598814351671061357088, −10.79628427239702374710276749576, −9.788214243165327798893074016281, −8.295807555393278150196326430706, −7.36999783250473151580649468740, −6.22624883629167588322663232187, −5.74506776298649855523462818506, −4.50006194537559584515945594938, −3.59679578519364468787988304450, −0.13647728657986140641126749360, 1.71648216094039351981449438118, 2.81744004257534451654719615190, 4.50868108136651230004061598863, 5.27172845754812765438308394727, 6.37194508033193581920161200436, 7.84218409821668274861285481799, 9.600429413130501225492094318061, 10.35112046239536590060637158776, 11.02313575251566678652011324070, 12.34092404007667981832458641315

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