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Properties

Label 2-15e2-9.4-c3-0-48
Degree $2$
Conductor $225$
Sign $-0.835 - 0.550i$
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  + (−1.54 − 2.67i)2-s + (4.77 − 2.05i)3-s + (−0.766 + 1.32i)4-s + (−12.8 − 9.58i)6-s + (−15.6 − 27.1i)7-s − 19.9·8-s + (18.5 − 19.6i)9-s + (9.59 + 16.6i)11-s + (−0.926 + 7.90i)12-s + (−10.4 + 18.1i)13-s + (−48.3 + 83.7i)14-s + (36.9 + 64.0i)16-s + 6.19·17-s + (−81.0 − 19.2i)18-s − 96.6·19-s + ⋯
L(s)  = 1  + (−0.545 − 0.945i)2-s + (0.918 − 0.395i)3-s + (−0.0957 + 0.165i)4-s + (−0.875 − 0.652i)6-s + (−0.845 − 1.46i)7-s − 0.882·8-s + (0.686 − 0.726i)9-s + (0.263 + 0.455i)11-s + (−0.0222 + 0.190i)12-s + (−0.223 + 0.387i)13-s + (−0.922 + 1.59i)14-s + (0.577 + 1.00i)16-s + 0.0883·17-s + (−1.06 − 0.252i)18-s − 1.16·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.835 - 0.550i$
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: \((2,\ 225,\ (\ :3/2),\ -0.835 - 0.550i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.334451 + 1.11578i\)
\(L(\frac12)\) \(\approx\) \(0.334451 + 1.11578i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.77 + 2.05i)T \)
5 \( 1 \)
good2 \( 1 + (1.54 + 2.67i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (15.6 + 27.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-9.59 - 16.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (10.4 - 18.1i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 6.19T + 4.91e3T^{2} \)
19 \( 1 + 96.6T + 6.85e3T^{2} \)
23 \( 1 + (-81.4 + 141. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (3.82 + 6.62i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (112. - 194. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 155.T + 5.06e4T^{2} \)
41 \( 1 + (-157. + 272. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (96.3 + 166. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-159. - 275. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 277.T + 1.48e5T^{2} \)
59 \( 1 + (-214. + 371. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-44.9 - 77.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-291. + 505. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 132.T + 3.57e5T^{2} \)
73 \( 1 + 259.T + 3.89e5T^{2} \)
79 \( 1 + (62.3 + 107. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-18.3 - 31.8i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 333.T + 7.04e5T^{2} \)
97 \( 1 + (514. + 891. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.87626472402740941912099474138, −10.27051334645293795384305380183, −9.399557287934761880105815524015, −8.580777528330638876296514509378, −7.14499038243436746797694981750, −6.54132089248832877417042498292, −4.22899991632710350306306777767, −3.17667193549366254664278765896, −1.85942851012641905814760813079, −0.48945360596334156190081022485, 2.49320296896381509939695550089, 3.50786872780663280919731970957, 5.41940152809358130836692980983, 6.39058420871984798972311895755, 7.57223058744819172088589710901, 8.563573336669856532843137928590, 9.118145152964830518460020968586, 9.870413697156077282486530829563, 11.39463268474080265209696945079, 12.55476711219157875315702213921

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