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Properties

Label 2-15e2-25.21-c1-0-5
Degree $2$
Conductor $225$
Sign $0.915 - 0.402i$
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  + (−0.167 + 0.515i)2-s + (1.38 + 1.00i)4-s + (1.16 − 1.91i)5-s + 1.08·7-s + (−1.62 + 1.18i)8-s + (0.789 + 0.918i)10-s + (0.900 − 2.77i)11-s + (0.298 + 0.918i)13-s + (−0.182 + 0.560i)14-s + (0.718 + 2.21i)16-s + (−2.15 + 1.56i)17-s + (1.59 − 1.15i)19-s + (3.52 − 1.47i)20-s + (1.27 + 0.928i)22-s + (−2.11 + 6.49i)23-s + ⋯
L(s)  = 1  + (−0.118 + 0.364i)2-s + (0.690 + 0.501i)4-s + (0.519 − 0.854i)5-s + 0.411·7-s + (−0.574 + 0.417i)8-s + (0.249 + 0.290i)10-s + (0.271 − 0.835i)11-s + (0.0828 + 0.254i)13-s + (−0.0487 + 0.149i)14-s + (0.179 + 0.552i)16-s + (−0.521 + 0.379i)17-s + (0.364 − 0.265i)19-s + (0.787 − 0.328i)20-s + (0.272 + 0.197i)22-s + (−0.440 + 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.915 - 0.402i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.915 - 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42206 + 0.299024i\)
\(L(\frac12)\) \(\approx\) \(1.42206 + 0.299024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.16 + 1.91i)T \)
good2 \( 1 + (0.167 - 0.515i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 - 1.08T + 7T^{2} \)
11 \( 1 + (-0.900 + 2.77i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.298 - 0.918i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.15 - 1.56i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.59 + 1.15i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.11 - 6.49i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.51 - 4.00i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.01 - 1.46i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.54 + 7.82i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.26 + 10.0i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.72T + 43T^{2} \)
47 \( 1 + (7.29 + 5.30i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.47 - 3.25i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.08 - 12.5i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.12 + 3.47i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.15 - 5.20i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (6.31 + 4.59i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.82 + 14.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.34 - 3.88i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.53 - 1.11i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.353 + 1.08i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.01 - 3.64i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.17438310357602535710433167535, −11.51851726775455834368870929851, −10.44683377314954447452798103649, −9.008060832262921839789678635865, −8.479394580398050855648312355503, −7.31085111513348218284987191845, −6.17584281296039315686564833376, −5.18854875440606023928787893428, −3.57088785803558858130449902199, −1.81774396711100185688136753000, 1.81862280819199131328737928357, 2.98977510088139629008516463177, 4.82866144473269530121121978102, 6.27930147792452839955135461048, 6.86710220461526821096513458904, 8.194785816449664841615788246310, 9.789703217422116321887058241044, 10.13185687625827618837805803324, 11.24624847098022078606077212986, 11.85230872426428003069051837871

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