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Properties

Label 2-15e2-9.7-c3-0-53
Degree $2$
Conductor $225$
Sign $-0.0412 - 0.999i$
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  + (2.52 − 4.37i)2-s + (−0.559 − 5.16i)3-s + (−8.77 − 15.1i)4-s + (−24.0 − 10.6i)6-s + (−10.5 + 18.1i)7-s − 48.2·8-s + (−26.3 + 5.78i)9-s + (14.2 − 24.7i)11-s + (−73.5 + 53.8i)12-s + (5.03 + 8.72i)13-s + (53.1 + 91.9i)14-s + (−51.7 + 89.6i)16-s − 82.7·17-s + (−41.3 + 130. i)18-s + 1.91·19-s + ⋯
L(s)  = 1  + (0.893 − 1.54i)2-s + (−0.107 − 0.994i)3-s + (−1.09 − 1.89i)4-s + (−1.63 − 0.721i)6-s + (−0.567 + 0.982i)7-s − 2.13·8-s + (−0.976 + 0.214i)9-s + (0.391 − 0.678i)11-s + (−1.77 + 1.29i)12-s + (0.107 + 0.186i)13-s + (1.01 + 1.75i)14-s + (−0.808 + 1.40i)16-s − 1.18·17-s + (−0.541 + 1.70i)18-s + 0.0231·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0412 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0412 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.0412 - 0.999i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.0412 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.998359 + 1.04047i\)
\(L(\frac12)\) \(\approx\) \(0.998359 + 1.04047i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.559 + 5.16i)T \)
5 \( 1 \)
good2 \( 1 + (-2.52 + 4.37i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (10.5 - 18.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-14.2 + 24.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-5.03 - 8.72i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 82.7T + 4.91e3T^{2} \)
19 \( 1 - 1.91T + 6.85e3T^{2} \)
23 \( 1 + (85.2 + 147. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-128. + 222. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (24.0 + 41.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 161.T + 5.06e4T^{2} \)
41 \( 1 + (139. + 241. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (134. - 233. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-4.79 + 8.30i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 35.7T + 1.48e5T^{2} \)
59 \( 1 + (-281. - 487. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-39.6 + 68.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (233. + 404. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 316.T + 3.57e5T^{2} \)
73 \( 1 + 633.T + 3.89e5T^{2} \)
79 \( 1 + (-395. + 685. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (114. - 197. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 53.9T + 7.04e5T^{2} \)
97 \( 1 + (-48.3 + 83.7i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.54938607407126128643773585935, −10.52100361119747531464846717706, −9.279315174719554407116884507853, −8.395052621635947935840463867351, −6.46116686032028566841752262936, −5.81058462073798626449352975902, −4.35557328175645683844305348852, −2.88226475728068602239425687232, −2.07299815344636748551190089354, −0.43460098585283283417499740510, 3.46026114339059494347770884923, 4.26774406377922841852837022154, 5.19515342600443004851572796331, 6.40526602488423613302190623216, 7.15098764276188494632474839240, 8.374040089764867711293825463306, 9.427882603144470120832394960847, 10.43729755898963053583390852816, 11.72018202130827801488100808889, 12.93256434764173585212628267412

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