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Properties

Label 2-15e2-9.4-c3-0-6
Degree $2$
Conductor $225$
Sign $0.816 - 0.577i$
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  + (−2.03 − 3.53i)2-s + (4.38 + 2.78i)3-s + (−4.30 + 7.46i)4-s + (0.877 − 21.1i)6-s + (6.67 + 11.5i)7-s + 2.51·8-s + (11.5 + 24.4i)9-s + (−5.78 − 10.0i)11-s + (−39.6 + 20.7i)12-s + (−20.0 + 34.7i)13-s + (27.1 − 47.1i)14-s + (29.3 + 50.8i)16-s − 93.3·17-s + (62.7 − 90.4i)18-s + 75.1·19-s + ⋯
L(s)  = 1  + (−0.720 − 1.24i)2-s + (0.844 + 0.535i)3-s + (−0.538 + 0.932i)4-s + (0.0597 − 1.43i)6-s + (0.360 + 0.623i)7-s + 0.110·8-s + (0.426 + 0.904i)9-s + (−0.158 − 0.274i)11-s + (−0.954 + 0.499i)12-s + (−0.427 + 0.740i)13-s + (0.519 − 0.899i)14-s + (0.458 + 0.794i)16-s − 1.33·17-s + (0.821 − 1.18i)18-s + 0.906·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.816 - 0.577i$
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: \((2,\ 225,\ (\ :3/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.15942 + 0.368364i\)
\(L(\frac12)\) \(\approx\) \(1.15942 + 0.368364i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.38 - 2.78i)T \)
5 \( 1 \)
good2 \( 1 + (2.03 + 3.53i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-6.67 - 11.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (5.78 + 10.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (20.0 - 34.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 93.3T + 4.91e3T^{2} \)
19 \( 1 - 75.1T + 6.85e3T^{2} \)
23 \( 1 + (71.0 - 122. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-87.0 - 150. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (124. - 215. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 82.9T + 5.06e4T^{2} \)
41 \( 1 + (-224. + 389. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-139. - 242. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-16.9 - 29.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 423.T + 1.48e5T^{2} \)
59 \( 1 + (-307. + 533. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-251. - 435. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (28.8 - 50.0i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 252.T + 3.57e5T^{2} \)
73 \( 1 + 823.T + 3.89e5T^{2} \)
79 \( 1 + (-102. - 178. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-323. - 559. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.32e3T + 7.04e5T^{2} \)
97 \( 1 + (281. + 488. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.61325494873405819859942780002, −10.85786618928042316967771806760, −9.905460159770509987285808543571, −9.024256646176161574453998874549, −8.595468474420889705037238420844, −7.22793002055647714849805648803, −5.34404646689713132464270148211, −3.91846262914305711931471779106, −2.70965316330345032238676868370, −1.72095122634539924241140404064, 0.56740528455588378456505135789, 2.53631505826113033349338385978, 4.31877483125622722854831514233, 5.93254937256414104322769931714, 7.03144418685419796422870333950, 7.68315691433309383534527380079, 8.420781047798185183587736455572, 9.393788892256412596095471984497, 10.29908037874272958682015494075, 11.78632321536283746489843607600

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