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Properties

Label 2-15e2-45.23-c1-0-10
Degree $2$
Conductor $225$
Sign $0.971 + 0.237i$
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  + (−0.0499 + 0.186i)2-s + (0.806 − 1.53i)3-s + (1.69 + 0.981i)4-s + (0.245 + 0.226i)6-s + (2.35 + 0.632i)7-s + (−0.540 + 0.540i)8-s + (−1.69 − 2.47i)9-s + (−2.14 + 1.23i)11-s + (2.87 − 1.81i)12-s + (−1.57 + 0.422i)13-s + (−0.235 + 0.407i)14-s + (1.88 + 3.27i)16-s + (0.403 + 0.403i)17-s + (0.545 − 0.193i)18-s − 4.28i·19-s + ⋯
L(s)  = 1  + (−0.0352 + 0.131i)2-s + (0.465 − 0.885i)3-s + (0.849 + 0.490i)4-s + (0.100 + 0.0925i)6-s + (0.891 + 0.238i)7-s + (−0.191 + 0.191i)8-s + (−0.566 − 0.823i)9-s + (−0.646 + 0.373i)11-s + (0.829 − 0.523i)12-s + (−0.436 + 0.117i)13-s + (−0.0629 + 0.108i)14-s + (0.472 + 0.818i)16-s + (0.0979 + 0.0979i)17-s + (0.128 − 0.0455i)18-s − 0.983i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.971 + 0.237i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.971 + 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62323 - 0.195970i\)
\(L(\frac12)\) \(\approx\) \(1.62323 - 0.195970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.806 + 1.53i)T \)
5 \( 1 \)
good2 \( 1 + (0.0499 - 0.186i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (-2.35 - 0.632i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (2.14 - 1.23i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.57 - 0.422i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.403 - 0.403i)T + 17iT^{2} \)
19 \( 1 + 4.28iT - 19T^{2} \)
23 \( 1 + (1.82 + 6.82i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.20 - 5.55i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.97 - 3.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.171 + 0.171i)T - 37iT^{2} \)
41 \( 1 + (6.52 + 3.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.32 - 4.95i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.780 - 2.91i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.12 - 6.12i)T - 53iT^{2} \)
59 \( 1 + (-2.27 + 3.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.235 + 0.408i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.443 - 1.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.50iT - 71T^{2} \)
73 \( 1 + (6.88 + 6.88i)T + 73iT^{2} \)
79 \( 1 + (-6.50 + 3.75i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.6 - 2.85i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.90T + 89T^{2} \)
97 \( 1 + (1.41 + 0.379i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.30587393709643030445870961922, −11.44973297632380393903234591079, −10.48052597950914625844010519380, −8.863474328116765502145413909386, −8.098676965365665567036222256731, −7.26815327302698981897974590524, −6.41193247333960357199991319570, −4.90462267882424870767927332485, −2.98225350498413520083731061042, −1.93655876452131087457040210859, 2.02997163679280467484374166194, 3.45236381591024704199432257631, 4.95145637108212669452364018885, 5.87310842173005029915120435566, 7.54368837583964353025611205039, 8.226100561788149117957428449200, 9.707108203039476493620385007661, 10.29058268029836775133718977814, 11.22606370587233991937008004329, 11.87981678781816878604583861588

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