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Properties

Label 2-15e2-45.2-c1-0-14
Degree $2$
Conductor $225$
Sign $-0.622 + 0.782i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.347 − 1.29i)2-s + (1.18 − 1.25i)3-s + (0.170 − 0.0981i)4-s + (−2.04 − 1.10i)6-s + (−1.97 + 0.530i)7-s + (−2.08 − 2.08i)8-s + (−0.170 − 2.99i)9-s + (−0.762 − 0.440i)11-s + (0.0786 − 0.330i)12-s + (5.36 + 1.43i)13-s + (1.37 + 2.38i)14-s + (−1.78 + 3.09i)16-s + (1.13 − 1.13i)17-s + (−3.82 + 1.26i)18-s + 1.52i·19-s + ⋯
L(s)  = 1  + (−0.245 − 0.917i)2-s + (0.686 − 0.726i)3-s + (0.0850 − 0.0490i)4-s + (−0.835 − 0.451i)6-s + (−0.747 + 0.200i)7-s + (−0.737 − 0.737i)8-s + (−0.0566 − 0.998i)9-s + (−0.229 − 0.132i)11-s + (0.0227 − 0.0955i)12-s + (1.48 + 0.398i)13-s + (0.367 + 0.636i)14-s + (−0.446 + 0.772i)16-s + (0.275 − 0.275i)17-s + (−0.901 + 0.297i)18-s + 0.349i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.622 + 0.782i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.622 + 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584540 - 1.21102i\)
\(L(\frac12)\) \(\approx\) \(0.584540 - 1.21102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 + 1.25i)T \)
5 \( 1 \)
good2 \( 1 + (0.347 + 1.29i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (1.97 - 0.530i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.762 + 0.440i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.36 - 1.43i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.13 + 1.13i)T - 17iT^{2} \)
19 \( 1 - 1.52iT - 19T^{2} \)
23 \( 1 + (0.410 - 1.53i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.796 + 1.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.49 - 6.05i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.25 - 4.25i)T + 37iT^{2} \)
41 \( 1 + (-3.11 + 1.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.497 + 1.85i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.14 - 7.99i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.65 + 4.65i)T + 53iT^{2} \)
59 \( 1 + (-3.81 - 6.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.64 + 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.859 + 3.20i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.89iT - 71T^{2} \)
73 \( 1 + (1.58 - 1.58i)T - 73iT^{2} \)
79 \( 1 + (6.69 + 3.86i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.59 + 2.57i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.62T + 89T^{2} \)
97 \( 1 + (3.82 - 1.02i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.92903433046678594999871895833, −11.04431712266723342137092335750, −9.910673276472410350037047495774, −9.122612873932928940148840014774, −8.144652509322406055217805581699, −6.73958760233655306331447181060, −6.03181131537428587320673427318, −3.69083341062415029374158400005, −2.77027998822008171798361644953, −1.28915196766123443165310179113, 2.75651697970226723848346136097, 3.93883888101376551029046684415, 5.56630472439304641022397931950, 6.55636649339409536430134194782, 7.78486101773809333620160411867, 8.507233091833844915332483903270, 9.441651740613497959050418278607, 10.49159447855346976273062687822, 11.41485666027504328477549618999, 12.87231194287040004363804599163

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