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Properties

Label 2-15e2-45.23-c3-0-23
Degree $2$
Conductor $225$
Sign $0.133 - 0.991i$
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  + (−1.11 + 4.14i)2-s + (4.87 − 1.80i)3-s + (−9.00 − 5.20i)4-s + (2.06 + 22.1i)6-s + (14.5 + 3.90i)7-s + (7.28 − 7.28i)8-s + (20.4 − 17.5i)9-s + (42.5 − 24.5i)11-s + (−53.2 − 9.09i)12-s + (33.7 − 9.04i)13-s + (−32.3 + 56.0i)14-s + (−19.5 − 33.7i)16-s + (−18.9 − 18.9i)17-s + (50.0 + 104. i)18-s − 53.7i·19-s + ⋯
L(s)  = 1  + (−0.392 + 1.46i)2-s + (0.937 − 0.347i)3-s + (−1.12 − 0.650i)4-s + (0.140 + 1.51i)6-s + (0.787 + 0.211i)7-s + (0.321 − 0.321i)8-s + (0.759 − 0.650i)9-s + (1.16 − 0.673i)11-s + (−1.28 − 0.218i)12-s + (0.720 − 0.193i)13-s + (−0.618 + 1.07i)14-s + (−0.304 − 0.527i)16-s + (−0.269 − 0.269i)17-s + (0.655 + 1.36i)18-s − 0.649i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.133 - 0.991i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.133 - 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.69824 + 1.48493i\)
\(L(\frac12)\) \(\approx\) \(1.69824 + 1.48493i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.87 + 1.80i)T \)
5 \( 1 \)
good2 \( 1 + (1.11 - 4.14i)T + (-6.92 - 4i)T^{2} \)
7 \( 1 + (-14.5 - 3.90i)T + (297. + 171.5i)T^{2} \)
11 \( 1 + (-42.5 + 24.5i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-33.7 + 9.04i)T + (1.90e3 - 1.09e3i)T^{2} \)
17 \( 1 + (18.9 + 18.9i)T + 4.91e3iT^{2} \)
19 \( 1 + 53.7iT - 6.85e3T^{2} \)
23 \( 1 + (-45.7 - 170. i)T + (-1.05e4 + 6.08e3i)T^{2} \)
29 \( 1 + (-110. - 190. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (22.1 - 38.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (169. - 169. i)T - 5.06e4iT^{2} \)
41 \( 1 + (-42.1 - 24.3i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-92.3 + 344. i)T + (-6.88e4 - 3.97e4i)T^{2} \)
47 \( 1 + (137. - 512. i)T + (-8.99e4 - 5.19e4i)T^{2} \)
53 \( 1 + (198. - 198. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-64.3 + 111. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (33.4 + 57.9i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (18.6 + 69.7i)T + (-2.60e5 + 1.50e5i)T^{2} \)
71 \( 1 + 1.03e3iT - 3.57e5T^{2} \)
73 \( 1 + (339. + 339. i)T + 3.89e5iT^{2} \)
79 \( 1 + (-297. + 171. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-841. - 225. i)T + (4.95e5 + 2.85e5i)T^{2} \)
89 \( 1 - 230.T + 7.04e5T^{2} \)
97 \( 1 + (553. + 148. i)T + (7.90e5 + 4.56e5i)T^{2} \)
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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

−12.07434848640559204681794482675, −11.03935001326752828138953529485, −9.307303109691392821055624155072, −8.849762811824092525394957374277, −8.037947029329494455324937230921, −7.09331440862059661870373574081, −6.24029624363073355189304658411, −4.91912389264253843866694155904, −3.31966678827964638026149592550, −1.31801408180833255130035765273, 1.31741309065690926440431207321, 2.32151408078469644818125283428, 3.79699341059779687584757142289, 4.46367738778193545845798605158, 6.63828435655781849581543411413, 8.157748126758512735315174913672, 8.851064473495174984710800420538, 9.766116319670004929473884467660, 10.56279785654063008451322911369, 11.41965683471676404903616100531

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