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Properties

Label 2-15e2-45.32-c3-0-42
Degree $2$
Conductor $225$
Sign $0.872 - 0.489i$
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  + (4.73 + 1.27i)2-s + (4.68 − 2.23i)3-s + (13.9 + 8.03i)4-s + (25.0 − 4.64i)6-s + (−5.69 + 21.2i)7-s + (28.0 + 28.0i)8-s + (16.9 − 20.9i)9-s + (30.9 − 17.8i)11-s + (83.2 + 6.54i)12-s + (−10.4 − 38.8i)13-s + (−53.9 + 93.5i)14-s + (32.9 + 57.0i)16-s + (−54.1 + 54.1i)17-s + (107. − 77.8i)18-s + 55.2i·19-s + ⋯
L(s)  = 1  + (1.67 + 0.449i)2-s + (0.902 − 0.430i)3-s + (1.74 + 1.00i)4-s + (1.70 − 0.316i)6-s + (−0.307 + 1.14i)7-s + (1.23 + 1.23i)8-s + (0.629 − 0.777i)9-s + (0.848 − 0.489i)11-s + (2.00 + 0.157i)12-s + (−0.221 − 0.828i)13-s + (−1.03 + 1.78i)14-s + (0.514 + 0.891i)16-s + (−0.772 + 0.772i)17-s + (1.40 − 1.02i)18-s + 0.667i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.872 - 0.489i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.872 - 0.489i)\)

Particular Values

\(L(2)\) \(\approx\) \(5.72088 + 1.49518i\)
\(L(\frac12)\) \(\approx\) \(5.72088 + 1.49518i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.68 + 2.23i)T \)
5 \( 1 \)
good2 \( 1 + (-4.73 - 1.27i)T + (6.92 + 4i)T^{2} \)
7 \( 1 + (5.69 - 21.2i)T + (-297. - 171.5i)T^{2} \)
11 \( 1 + (-30.9 + 17.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (10.4 + 38.8i)T + (-1.90e3 + 1.09e3i)T^{2} \)
17 \( 1 + (54.1 - 54.1i)T - 4.91e3iT^{2} \)
19 \( 1 - 55.2iT - 6.85e3T^{2} \)
23 \( 1 + (96.6 - 25.9i)T + (1.05e4 - 6.08e3i)T^{2} \)
29 \( 1 + (74.2 + 128. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-102. + 177. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-151. - 151. i)T + 5.06e4iT^{2} \)
41 \( 1 + (38.4 + 22.1i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (42.2 + 11.3i)T + (6.88e4 + 3.97e4i)T^{2} \)
47 \( 1 + (473. + 126. i)T + (8.99e4 + 5.19e4i)T^{2} \)
53 \( 1 + (-215. - 215. i)T + 1.48e5iT^{2} \)
59 \( 1 + (21.8 - 37.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (16.1 + 28.0i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (1.00e3 - 268. i)T + (2.60e5 - 1.50e5i)T^{2} \)
71 \( 1 - 53.5iT - 3.57e5T^{2} \)
73 \( 1 + (23.7 - 23.7i)T - 3.89e5iT^{2} \)
79 \( 1 + (189. - 109. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (73.8 - 275. i)T + (-4.95e5 - 2.85e5i)T^{2} \)
89 \( 1 - 870.T + 7.04e5T^{2} \)
97 \( 1 + (-34.2 + 127. i)T + (-7.90e5 - 4.56e5i)T^{2} \)
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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

−12.20452219192589031766112082064, −11.59401079831620282797435938982, −9.829603338753143770368826448313, −8.612902570750084299628695465547, −7.70666415805373032620173914305, −6.32381492038966312397967709421, −5.88495905636466156821372873610, −4.25117876968549592553853592282, −3.25624647202099778542496325769, −2.16216620754891786843130487375, 1.85057484526185457448717456759, 3.18953415199220782685721169483, 4.21844419040305807482148795967, 4.74843951179776323346346964136, 6.58469800477858273653299653918, 7.25931178765678241249870926690, 8.993228333221941273976267731132, 10.00055152458204480860805398886, 10.96535204219887363854424018963, 11.88437735406351692285430092111

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