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Properties

Label 2-15e2-25.6-c1-0-5
Degree $2$
Conductor $225$
Sign $0.535 - 0.844i$
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  + (0.5 + 1.53i)2-s + (−0.5 + 0.363i)4-s + (0.690 − 2.12i)5-s + 0.618·7-s + (1.80 + 1.31i)8-s + 3.61·10-s + (1.61 + 4.97i)11-s + (0.572 − 1.76i)13-s + (0.309 + 0.951i)14-s + (−1.50 + 4.61i)16-s + (−4.23 − 3.07i)17-s + (−0.690 − 0.502i)19-s + (0.427 + 1.31i)20-s + (−6.85 + 4.97i)22-s + (−1.16 − 3.57i)23-s + ⋯
L(s)  = 1  + (0.353 + 1.08i)2-s + (−0.250 + 0.181i)4-s + (0.309 − 0.951i)5-s + 0.233·7-s + (0.639 + 0.464i)8-s + 1.14·10-s + (0.487 + 1.50i)11-s + (0.158 − 0.489i)13-s + (0.0825 + 0.254i)14-s + (−0.375 + 1.15i)16-s + (−1.02 − 0.746i)17-s + (−0.158 − 0.115i)19-s + (0.0954 + 0.293i)20-s + (−1.46 + 1.06i)22-s + (−0.242 − 0.746i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.535 - 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45623 + 0.800569i\)
\(L(\frac12)\) \(\approx\) \(1.45623 + 0.800569i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.690 + 2.12i)T \)
good2 \( 1 + (-0.5 - 1.53i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 - 0.618T + 7T^{2} \)
11 \( 1 + (-1.61 - 4.97i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.572 + 1.76i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.23 + 3.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.690 + 0.502i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.16 + 3.57i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.92 - 2.12i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.42 - 1.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.0729 - 0.224i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.236 + 0.726i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.85T + 43T^{2} \)
47 \( 1 + (-0.5 + 0.363i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.80 - 2.04i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.35 + 10.3i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.69 - 8.28i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.85 + 2.80i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.35 - 3.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.78 + 8.55i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.54 + 4.75i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.04 + 3.66i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.76 - 8.50i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.11 + 2.26i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.66057714993738833404708833558, −11.60766805620164478269859350032, −10.35191939507233200398585492902, −9.251094868300608055779950772959, −8.282364203319938659883156442403, −7.21597218966254580881205878998, −6.31247644315263619326520085801, −5.04019110117545233045493522709, −4.45854717219692928751856551430, −1.92366331863080516103889970660, 1.84206524063358666703503926607, 3.18342651284219635373576048364, 4.12961664661698091129487997365, 5.92367509451255423554146422096, 6.87077533599086226292170746110, 8.222942394282048637108043109441, 9.482694106766630820956880242130, 10.54699749334950731708751229032, 11.27824441084538617321711242723, 11.68461145801939377156686363429

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