Properties

Label 2-15e2-25.21-c3-0-1
Degree $2$
Conductor $225$
Sign $-0.620 + 0.783i$
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  + (−1.20 + 3.70i)2-s + (−5.80 − 4.22i)4-s + (3.34 − 10.6i)5-s + 6.27·7-s + (−2.58 + 1.87i)8-s + (35.5 + 25.2i)10-s + (−13.6 + 41.9i)11-s + (−13.2 − 40.7i)13-s + (−7.56 + 23.2i)14-s + (−21.5 − 66.4i)16-s + (−81.6 + 59.3i)17-s + (−65.1 + 47.3i)19-s + (−64.4 + 47.8i)20-s + (−139. − 101. i)22-s + (−48.5 + 149. i)23-s + ⋯
L(s)  = 1  + (−0.425 + 1.31i)2-s + (−0.726 − 0.527i)4-s + (0.298 − 0.954i)5-s + 0.339·7-s + (−0.114 + 0.0829i)8-s + (1.12 + 0.797i)10-s + (−0.373 + 1.15i)11-s + (−0.282 − 0.870i)13-s + (−0.144 + 0.444i)14-s + (−0.337 − 1.03i)16-s + (−1.16 + 0.846i)17-s + (−0.786 + 0.571i)19-s + (−0.720 + 0.535i)20-s + (−1.34 − 0.979i)22-s + (−0.440 + 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.620 + 0.783i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.620 + 0.783i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.149883 - 0.309883i\)
\(L(\frac12)\) \(\approx\) \(0.149883 - 0.309883i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.34 + 10.6i)T \)
good2 \( 1 + (1.20 - 3.70i)T + (-6.47 - 4.70i)T^{2} \)
7 \( 1 - 6.27T + 343T^{2} \)
11 \( 1 + (13.6 - 41.9i)T + (-1.07e3 - 782. i)T^{2} \)
13 \( 1 + (13.2 + 40.7i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (81.6 - 59.3i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (65.1 - 47.3i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (48.5 - 149. i)T + (-9.84e3 - 7.15e3i)T^{2} \)
29 \( 1 + (199. + 145. i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-69.6 + 50.6i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-61.0 - 188. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-17.2 - 53.1i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 211.T + 7.95e4T^{2} \)
47 \( 1 + (-85.9 - 62.4i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (153. + 111. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (140. + 432. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-171. + 528. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + (773. - 561. i)T + (9.29e4 - 2.86e5i)T^{2} \)
71 \( 1 + (-454. - 330. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (54.7 - 168. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (769. + 558. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-11.3 + 8.23i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + (143. - 440. i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (-1.05e3 - 764. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−12.66798654445565229810405909372, −11.50516840660108787315439553964, −10.06193808525621511002714474955, −9.271171949453616768629540205708, −8.154571821335482287187004284755, −7.68953103038469668422808840271, −6.31024890529096174247634470329, −5.41610515333012203408803140869, −4.36829528376315674317632899688, −1.98953822201703979465081796429, 0.14996463020033908168475125782, 2.06419912976957575392003709363, 2.91801937385713314027686635632, 4.33255443362769863010070844765, 6.08966618799291643637321052546, 7.09270992710745472257960212797, 8.662830384321041483571401466747, 9.352846646001680655912685683467, 10.64987766713945145931582492957, 10.95569157770816704611119275738

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