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Properties

Label 2-15e2-25.21-c1-0-2
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

\(L(1)\) \(\approx\) \(1.42149 + 0.298903i\)
\(L(\frac12)\) \(\approx\) \(1.42149 + 0.298903i\)
\(L(\frac{3}{2})\) 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

−12.17531882746266400906242154312, −11.30311764173593554880877731134, −10.72474777395921969028776089990, −9.609532887365882556795889667092, −8.026986817265425012875704853493, −7.35854335272976701577834570795, −6.43779734663103922664176578432, −4.67211547795430209255162870956, −3.40220455279282279370913031340, −2.20566925660868365729496243891, 1.42983693016397954418139490119, 3.49331128554454642804183873245, 5.14973037234227306340817909997, 5.75607604364043995620845180576, 7.29220672406547107329622142953, 8.032558969890887991892808250637, 9.112754104239991095829638796436, 10.41264652228567589886441481085, 11.30596421121484131982837167419, 12.02240250871811857872293165694

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