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Properties

Label 2-15e2-45.23-c3-0-4
Degree $2$
Conductor $225$
Sign $0.171 + 0.985i$
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  + (−1.15 + 4.29i)2-s + (2.52 + 4.54i)3-s + (−10.2 − 5.90i)4-s + (−22.4 + 5.62i)6-s + (7.13 + 1.91i)7-s + (11.9 − 11.9i)8-s + (−14.2 + 22.9i)9-s + (−7.55 + 4.36i)11-s + (0.999 − 61.3i)12-s + (−73.7 + 19.7i)13-s + (−16.4 + 28.4i)14-s + (−9.50 − 16.4i)16-s + (5.92 + 5.92i)17-s + (−82.1 − 87.6i)18-s − 106. i·19-s + ⋯
L(s)  = 1  + (−0.407 + 1.51i)2-s + (0.485 + 0.874i)3-s + (−1.27 − 0.738i)4-s + (−1.52 + 0.382i)6-s + (0.385 + 0.103i)7-s + (0.529 − 0.529i)8-s + (−0.527 + 0.849i)9-s + (−0.207 + 0.119i)11-s + (0.0240 − 1.47i)12-s + (−1.57 + 0.421i)13-s + (−0.313 + 0.543i)14-s + (−0.148 − 0.257i)16-s + (0.0844 + 0.0844i)17-s + (−1.07 − 1.14i)18-s − 1.28i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.171 + 0.985i$
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: \((2,\ 225,\ (\ :3/2),\ 0.171 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.544005 - 0.457677i\)
\(L(\frac12)\) \(\approx\) \(0.544005 - 0.457677i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.52 - 4.54i)T \)
5 \( 1 \)
good2 \( 1 + (1.15 - 4.29i)T + (-6.92 - 4i)T^{2} \)
7 \( 1 + (-7.13 - 1.91i)T + (297. + 171.5i)T^{2} \)
11 \( 1 + (7.55 - 4.36i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (73.7 - 19.7i)T + (1.90e3 - 1.09e3i)T^{2} \)
17 \( 1 + (-5.92 - 5.92i)T + 4.91e3iT^{2} \)
19 \( 1 + 106. iT - 6.85e3T^{2} \)
23 \( 1 + (27.4 + 102. i)T + (-1.05e4 + 6.08e3i)T^{2} \)
29 \( 1 + (-85.3 - 147. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (157. - 273. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-31.7 + 31.7i)T - 5.06e4iT^{2} \)
41 \( 1 + (-298. - 172. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (115. - 431. i)T + (-6.88e4 - 3.97e4i)T^{2} \)
47 \( 1 + (-60.9 + 227. i)T + (-8.99e4 - 5.19e4i)T^{2} \)
53 \( 1 + (-60.5 + 60.5i)T - 1.48e5iT^{2} \)
59 \( 1 + (25.4 - 44.0i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (36.1 + 62.5i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (4.59 + 17.1i)T + (-2.60e5 + 1.50e5i)T^{2} \)
71 \( 1 - 109. iT - 3.57e5T^{2} \)
73 \( 1 + (-144. - 144. i)T + 3.89e5iT^{2} \)
79 \( 1 + (140. - 81.1i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (279. + 74.9i)T + (4.95e5 + 2.85e5i)T^{2} \)
89 \( 1 + 1.10e3T + 7.04e5T^{2} \)
97 \( 1 + (-197. - 52.9i)T + (7.90e5 + 4.56e5i)T^{2} \)
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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

−12.74143709830125456113857270551, −11.37769193151538504290910162562, −10.19806706103122441183364518449, −9.287012928449418736012435146908, −8.577813313414603228748521485859, −7.60158108993522176739410259256, −6.70348559857568915198274251677, −5.15718018883365311003383515261, −4.66704123094707690399724428687, −2.66355766846472063380455145073, 0.30125804146869064015858570879, 1.77477986029154221662101296623, 2.72059851614991724510691604957, 4.00252785241781519816732893799, 5.75895068453205597503434499297, 7.44149885338311214318212963551, 8.127798874688327635580238283347, 9.387699150510118355372032772177, 10.03015086042532056711700121129, 11.20908614899631483733763790176

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