Properties

 Label 2-15e2-9.7-c3-0-12 Degree $2$ Conductor $225$ Sign $-0.820 + 0.571i$ Analytic cond. $13.2754$ Root an. cond. $3.64354$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−2.42 + 4.20i)2-s + (4.79 + 1.99i)3-s + (−7.76 − 13.4i)4-s + (−20.0 + 15.3i)6-s + (8.11 − 14.0i)7-s + 36.5·8-s + (19.0 + 19.1i)9-s + (−29.7 + 51.5i)11-s + (−10.4 − 80.0i)12-s + (13.5 + 23.4i)13-s + (39.3 + 68.2i)14-s + (−26.5 + 45.9i)16-s − 78.9·17-s + (−126. + 33.5i)18-s − 142.·19-s + ⋯
 L(s)  = 1 + (−0.857 + 1.48i)2-s + (0.923 + 0.384i)3-s + (−0.970 − 1.68i)4-s + (−1.36 + 1.04i)6-s + (0.438 − 0.759i)7-s + 1.61·8-s + (0.704 + 0.709i)9-s + (−0.815 + 1.41i)11-s + (−0.250 − 1.92i)12-s + (0.288 + 0.500i)13-s + (0.751 + 1.30i)14-s + (−0.414 + 0.718i)16-s − 1.12·17-s + (−1.65 + 0.438i)18-s − 1.71·19-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $-0.820 + 0.571i$ 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: $$(2,\ 225,\ (\ :3/2),\ -0.820 + 0.571i)$$

Particular Values

 $$L(2)$$ $$\approx$$ $$0.281734 - 0.898396i$$ $$L(\frac12)$$ $$\approx$$ $$0.281734 - 0.898396i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-4.79 - 1.99i)T$$
5 $$1$$
good2 $$1 + (2.42 - 4.20i)T + (-4 - 6.92i)T^{2}$$
7 $$1 + (-8.11 + 14.0i)T + (-171.5 - 297. i)T^{2}$$
11 $$1 + (29.7 - 51.5i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + (-13.5 - 23.4i)T + (-1.09e3 + 1.90e3i)T^{2}$$
17 $$1 + 78.9T + 4.91e3T^{2}$$
19 $$1 + 142.T + 6.85e3T^{2}$$
23 $$1 + (-42.8 - 74.1i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (113. - 195. i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (-75.3 - 130. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 + 234.T + 5.06e4T^{2}$$
41 $$1 + (36.9 + 63.9i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (-133. + 231. i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (-133. + 230. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 - 603.T + 1.48e5T^{2}$$
59 $$1 + (254. + 440. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (39.1 - 67.8i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (-244. - 422. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 73.2T + 3.57e5T^{2}$$
73 $$1 + 115.T + 3.89e5T^{2}$$
79 $$1 + (391. - 677. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (135. - 235. i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 - 211.T + 7.04e5T^{2}$$
97 $$1 + (-833. + 1.44e3i)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

−12.75656697691346308482158517236, −10.74757084809137872153469606382, −10.17640079450224714941599922269, −9.007589684300379583393752480582, −8.480389298607448080082135050025, −7.29084680636839840358470698842, −6.93310052170580197135587647337, −5.11631288032266871026426171865, −4.19642015968598757577254235414, −1.91795180402368999802200594611, 0.44294632581066993103611380186, 2.12057180890417972661154279424, 2.81703407247886075907347927039, 4.15630848796590584041130252145, 6.12545501427319263204410199993, 7.942921821296019193657503252965, 8.561667652998618181007123675445, 9.061914426181493136622735940108, 10.41187268167782449849524447099, 11.05391317044230406301646371173