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Properties

Label 2-15e2-45.32-c3-0-21
Degree $2$
Conductor $225$
Sign $0.355 + 0.934i$
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  + (−2.15 − 0.576i)2-s + (−4.75 − 2.09i)3-s + (−2.62 − 1.51i)4-s + (9.02 + 7.24i)6-s + (3.70 − 13.8i)7-s + (17.3 + 17.3i)8-s + (18.2 + 19.9i)9-s + (−35.0 + 20.2i)11-s + (9.32 + 12.7i)12-s + (14.4 + 53.7i)13-s + (−15.9 + 27.5i)14-s + (−15.2 − 26.4i)16-s + (−18.3 + 18.3i)17-s + (−27.7 − 53.3i)18-s − 29.3i·19-s + ⋯
L(s)  = 1  + (−0.760 − 0.203i)2-s + (−0.915 − 0.402i)3-s + (−0.328 − 0.189i)4-s + (0.614 + 0.492i)6-s + (0.199 − 0.745i)7-s + (0.768 + 0.768i)8-s + (0.675 + 0.737i)9-s + (−0.961 + 0.555i)11-s + (0.224 + 0.305i)12-s + (0.307 + 1.14i)13-s + (−0.304 + 0.526i)14-s + (−0.238 − 0.412i)16-s + (−0.262 + 0.262i)17-s + (−0.363 − 0.698i)18-s − 0.354i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.355 + 0.934i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.355 + 0.934i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.509885 - 0.351759i\)
\(L(\frac12)\) \(\approx\) \(0.509885 - 0.351759i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.75 + 2.09i)T \)
5 \( 1 \)
good2 \( 1 + (2.15 + 0.576i)T + (6.92 + 4i)T^{2} \)
7 \( 1 + (-3.70 + 13.8i)T + (-297. - 171.5i)T^{2} \)
11 \( 1 + (35.0 - 20.2i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-14.4 - 53.7i)T + (-1.90e3 + 1.09e3i)T^{2} \)
17 \( 1 + (18.3 - 18.3i)T - 4.91e3iT^{2} \)
19 \( 1 + 29.3iT - 6.85e3T^{2} \)
23 \( 1 + (-176. + 47.3i)T + (1.05e4 - 6.08e3i)T^{2} \)
29 \( 1 + (-39.7 - 68.8i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-62.6 + 108. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (248. + 248. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-155. - 89.6i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (361. + 96.7i)T + (6.88e4 + 3.97e4i)T^{2} \)
47 \( 1 + (-253. - 67.7i)T + (8.99e4 + 5.19e4i)T^{2} \)
53 \( 1 + (-56.9 - 56.9i)T + 1.48e5iT^{2} \)
59 \( 1 + (-213. + 370. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (400. + 693. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-576. + 154. i)T + (2.60e5 - 1.50e5i)T^{2} \)
71 \( 1 - 655. iT - 3.57e5T^{2} \)
73 \( 1 + (-91.3 + 91.3i)T - 3.89e5iT^{2} \)
79 \( 1 + (-529. + 305. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-55.3 + 206. i)T + (-4.95e5 - 2.85e5i)T^{2} \)
89 \( 1 + 451.T + 7.04e5T^{2} \)
97 \( 1 + (310. - 1.15e3i)T + (-7.90e5 - 4.56e5i)T^{2} \)
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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

−11.15466385871251588693695134686, −10.78708938585980446712901697346, −9.830011822148819093822298109404, −8.723529353003908961351049577451, −7.55787008244758839639068875467, −6.74148854680412646754973914504, −5.20848193785152107635122273591, −4.41366725815927791645619219219, −1.92312599718406412709374391358, −0.59342595156048589954730811108, 0.819058116583339274687043038235, 3.24624971353528735188765464447, 4.87351535928150673385425196517, 5.65235922825646911796954227428, 7.04138542166140389939052671499, 8.217216299831858500290877276681, 8.980649259827324119110502151850, 10.14741235775599326861602320959, 10.73561755602639417316534957668, 11.86180939513941392142230731270

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