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Properties

Label 2-15e2-9.7-c3-0-44
Degree $2$
Conductor $225$
Sign $-0.835 + 0.550i$
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  + (1.54 − 2.67i)2-s + (−4.77 − 2.05i)3-s + (−0.766 − 1.32i)4-s + (−12.8 + 9.58i)6-s + (15.6 − 27.1i)7-s + 19.9·8-s + (18.5 + 19.6i)9-s + (9.59 − 16.6i)11-s + (0.926 + 7.90i)12-s + (10.4 + 18.1i)13-s + (−48.3 − 83.7i)14-s + (36.9 − 64.0i)16-s − 6.19·17-s + (81.0 − 19.2i)18-s − 96.6·19-s + ⋯
L(s)  = 1  + (0.545 − 0.945i)2-s + (−0.918 − 0.395i)3-s + (−0.0957 − 0.165i)4-s + (−0.875 + 0.652i)6-s + (0.845 − 1.46i)7-s + 0.882·8-s + (0.686 + 0.726i)9-s + (0.263 − 0.455i)11-s + (0.0222 + 0.190i)12-s + (0.223 + 0.387i)13-s + (−0.922 − 1.59i)14-s + (0.577 − 1.00i)16-s − 0.0883·17-s + (1.06 − 0.252i)18-s − 1.16·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.550i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.835 + 0.550i$
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: \((2,\ 225,\ (\ :3/2),\ -0.835 + 0.550i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.561897 - 1.87457i\)
\(L(\frac12)\) \(\approx\) \(0.561897 - 1.87457i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.77 + 2.05i)T \)
5 \( 1 \)
good2 \( 1 + (-1.54 + 2.67i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-15.6 + 27.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-9.59 + 16.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-10.4 - 18.1i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 6.19T + 4.91e3T^{2} \)
19 \( 1 + 96.6T + 6.85e3T^{2} \)
23 \( 1 + (81.4 + 141. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (3.82 - 6.62i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (112. + 194. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 155.T + 5.06e4T^{2} \)
41 \( 1 + (-157. - 272. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-96.3 + 166. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (159. - 275. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 277.T + 1.48e5T^{2} \)
59 \( 1 + (-214. - 371. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-44.9 + 77.8i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (291. + 505. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 132.T + 3.57e5T^{2} \)
73 \( 1 - 259.T + 3.89e5T^{2} \)
79 \( 1 + (62.3 - 107. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (18.3 - 31.8i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 333.T + 7.04e5T^{2} \)
97 \( 1 + (-514. + 891. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.19316860828205669708747703226, −10.98171166322318002969985928220, −10.07679030034865410093301860041, −8.183930575409930293068336688646, −7.29752167851589257296181864115, −6.19427423649632547262303859046, −4.56255364516082727602610686368, −4.06519013889162681939733923288, −2.04816683281175284595353242569, −0.799510867230065132731314869375, 1.79454886144428570731764420917, 4.14567251596707340507943936275, 5.25587547696011776092255687919, 5.77001219867224725021151265587, 6.79213640037048874306703607174, 8.026583230767396691260605554386, 9.182708402101623953621564673989, 10.42450999395087442253292972946, 11.32088006542467555437978733863, 12.17360616711110884871845012930

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