Properties

Label 2-15e2-25.11-c3-0-22
Degree $2$
Conductor $225$
Sign $0.194 + 0.980i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.772 + 0.561i)2-s + (−2.19 + 6.74i)4-s + (−11.0 − 1.36i)5-s + 12.4·7-s + (−4.45 − 13.7i)8-s + (9.34 − 5.17i)10-s + (−36.1 + 26.2i)11-s + (5.77 + 4.19i)13-s + (−9.64 + 7.00i)14-s + (−34.7 − 25.2i)16-s + (−8.19 − 25.2i)17-s + (−14.3 − 44.1i)19-s + (33.4 − 71.8i)20-s + (13.2 − 40.6i)22-s + (117. − 85.4i)23-s + ⋯
L(s)  = 1  + (−0.273 + 0.198i)2-s + (−0.273 + 0.842i)4-s + (−0.992 − 0.121i)5-s + 0.674·7-s + (−0.196 − 0.605i)8-s + (0.295 − 0.163i)10-s + (−0.991 + 0.720i)11-s + (0.123 + 0.0894i)13-s + (−0.184 + 0.133i)14-s + (−0.542 − 0.394i)16-s + (−0.116 − 0.359i)17-s + (−0.173 − 0.532i)19-s + (0.374 − 0.802i)20-s + (0.127 − 0.393i)22-s + (1.06 − 0.775i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.194 + 0.980i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.194 + 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.414811 - 0.340707i\)
\(L(\frac12)\) \(\approx\) \(0.414811 - 0.340707i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (11.0 + 1.36i)T \)
good2 \( 1 + (0.772 - 0.561i)T + (2.47 - 7.60i)T^{2} \)
7 \( 1 - 12.4T + 343T^{2} \)
11 \( 1 + (36.1 - 26.2i)T + (411. - 1.26e3i)T^{2} \)
13 \( 1 + (-5.77 - 4.19i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (8.19 + 25.2i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (14.3 + 44.1i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 + (-117. + 85.4i)T + (3.75e3 - 1.15e4i)T^{2} \)
29 \( 1 + (-65.9 + 202. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (45.7 + 140. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (325. + 236. i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (-189. - 137. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 87.5T + 7.95e4T^{2} \)
47 \( 1 + (-23.4 + 72.2i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (51.6 - 158. i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (481. + 349. i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (-700. + 508. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + (125. + 386. i)T + (-2.43e5 + 1.76e5i)T^{2} \)
71 \( 1 + (162. - 500. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (810. - 588. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (41.3 - 127. i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (159. + 491. i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + (494. - 358. i)T + (2.17e5 - 6.70e5i)T^{2} \)
97 \( 1 + (506. - 1.55e3i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.61572572746988035481281232028, −10.78471277368549737240487768530, −9.389988761721365028674637134517, −8.389577433069772679439711267043, −7.71981188147373931640208224520, −6.91789173798483666514846255121, −4.96991353304327481228399499260, −4.14700704904629098165243374278, −2.66764836370677833630888216609, −0.26791698306881162624111726396, 1.30706147925499375927632352840, 3.17325497448269212319648500589, 4.72557184708144073732318697963, 5.60892689883364874971045937140, 7.10531379846958693779282382045, 8.273909475534934751851181265726, 8.873373814462581972049875274255, 10.44264979808940050668313068320, 10.84055421250375332403081259659, 11.74454536468285580008133400402

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