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Properties

Label 2-15e2-25.9-c1-0-3
Degree $2$
Conductor $225$
Sign $0.233 - 0.972i$
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  + (−1.49 + 2.05i)2-s + (−1.37 − 4.22i)4-s + (−0.227 − 2.22i)5-s + 1.04i·7-s + (5.88 + 1.91i)8-s + (4.90 + 2.85i)10-s + (2.40 + 1.74i)11-s + (3.33 + 4.58i)13-s + (−2.13 − 1.55i)14-s + (−5.52 + 4.01i)16-s + (4.83 + 1.57i)17-s + (1.65 − 5.10i)19-s + (−9.08 + 4.01i)20-s + (−7.17 + 2.32i)22-s + (2.26 − 3.12i)23-s + ⋯
L(s)  = 1  + (−1.05 + 1.45i)2-s + (−0.685 − 2.11i)4-s + (−0.101 − 0.994i)5-s + 0.393i·7-s + (2.08 + 0.676i)8-s + (1.55 + 0.901i)10-s + (0.724 + 0.526i)11-s + (0.924 + 1.27i)13-s + (−0.570 − 0.414i)14-s + (−1.38 + 1.00i)16-s + (1.17 + 0.381i)17-s + (0.380 − 1.17i)19-s + (−2.03 + 0.897i)20-s + (−1.52 + 0.496i)22-s + (0.473 − 0.651i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.233 - 0.972i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.233 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.580948 + 0.457836i\)
\(L(\frac12)\) \(\approx\) \(0.580948 + 0.457836i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.227 + 2.22i)T \)
good2 \( 1 + (1.49 - 2.05i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 - 1.04iT - 7T^{2} \)
11 \( 1 + (-2.40 - 1.74i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.33 - 4.58i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.83 - 1.57i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.65 + 5.10i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.26 + 3.12i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.210 - 0.646i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.262 + 0.808i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.950 - 1.30i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.942 + 0.684i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.68iT - 43T^{2} \)
47 \( 1 + (-3.12 + 1.01i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (12.0 - 3.92i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.59 - 1.88i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.38 - 3.18i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.883 + 0.287i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.436 - 1.34i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.65 - 9.16i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.447 - 1.37i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (10.9 + 3.54i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.33 + 5.33i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.73 + 1.86i)T + (78.4 - 57.0i)T^{2} \)
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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

−12.44871837873314153783068167154, −11.39834344576028144834677564824, −9.966003862603043120147248087366, −9.009220699514601920562921014628, −8.734586426887151182651763231066, −7.50027099060144270824094912189, −6.53626748062430513228463494577, −5.52166272577275524211217802905, −4.36357951199877890645640602403, −1.28117190032573051301860508189, 1.20094681855193157520430589917, 3.11369323489304331660145768189, 3.65684294020175865985999790231, 5.94130082329378033774179439541, 7.53066511623464470171361834836, 8.202590578243720904043223399183, 9.491511757194075603579690959127, 10.24858633370537560693853312552, 10.96452881879879386248194366629, 11.65955307026905546124772360273

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