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Properties

Label 2-15e2-45.4-c3-0-1
Degree $2$
Conductor $225$
Sign $-0.884 - 0.467i$
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  + (3.69 − 2.13i)2-s + (−2.76 + 4.39i)3-s + (5.08 − 8.80i)4-s + (−0.854 + 22.1i)6-s + (−26.6 + 15.3i)7-s − 9.21i·8-s + (−11.6 − 24.3i)9-s + (−20.3 − 35.2i)11-s + (24.6 + 46.7i)12-s + (−54.7 − 31.6i)13-s + (−65.5 + 113. i)14-s + (21.0 + 36.3i)16-s + 6.58i·17-s + (−94.9 − 65.0i)18-s − 75.3·19-s + ⋯
L(s)  = 1  + (1.30 − 0.753i)2-s + (−0.533 + 0.846i)3-s + (0.635 − 1.10i)4-s + (−0.0581 + 1.50i)6-s + (−1.43 + 0.830i)7-s − 0.407i·8-s + (−0.431 − 0.902i)9-s + (−0.557 − 0.966i)11-s + (0.592 + 1.12i)12-s + (−1.16 − 0.674i)13-s + (−1.25 + 2.16i)14-s + (0.328 + 0.568i)16-s + 0.0940i·17-s + (−1.24 − 0.851i)18-s − 0.910·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.884 - 0.467i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.884 - 0.467i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0785121 + 0.316571i\)
\(L(\frac12)\) \(\approx\) \(0.0785121 + 0.316571i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.76 - 4.39i)T \)
5 \( 1 \)
good2 \( 1 + (-3.69 + 2.13i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (26.6 - 15.3i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (20.3 + 35.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (54.7 + 31.6i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 6.58iT - 4.91e3T^{2} \)
19 \( 1 + 75.3T + 6.85e3T^{2} \)
23 \( 1 + (-54.0 - 31.1i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-24.8 - 42.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (51.5 - 89.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 282. iT - 5.06e4T^{2} \)
41 \( 1 + (-78.7 + 136. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (292. - 168. i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (38.5 - 22.2i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 26.2iT - 1.48e5T^{2} \)
59 \( 1 + (212. - 368. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (425. + 736. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-83.4 - 48.1i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 952.T + 3.57e5T^{2} \)
73 \( 1 + 50.8iT - 3.89e5T^{2} \)
79 \( 1 + (98.6 + 170. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (171. - 98.8i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + (-1.23e3 + 715. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.45202646596967498870266061391, −11.37923269012666319063522636952, −10.50791095043029539356104841754, −9.712694870453166128221973306975, −8.531189075485599969996911572071, −6.48065210085066142635361070468, −5.64263628739304028673872403587, −4.84654487552187064422725070305, −3.38696465461594093187140976283, −2.79200357340458232763163584528, 0.085108309097334837097628827739, 2.53413096234800428543297468555, 4.12861013044131617308157854172, 5.14571733200058101515612811059, 6.39250687515442183172753009748, 6.95727912664359870243644450922, 7.63394901327848450966221168065, 9.562264143506864045225983755272, 10.52951242497998110494856383055, 11.97898214043770819453076335830

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