# Properties

 Label 2-15e2-9.7-c3-0-27 Degree $2$ Conductor $225$ Sign $0.948 - 0.315i$ Analytic cond. $13.2754$ Root an. cond. $3.64354$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (1.07 − 1.85i)2-s + (−2.66 + 4.46i)3-s + (1.69 + 2.94i)4-s + (5.43 + 9.73i)6-s + (11.6 − 20.1i)7-s + 24.4·8-s + (−12.8 − 23.7i)9-s + (−6.24 + 10.8i)11-s + (−17.6 − 0.249i)12-s + (25.3 + 43.8i)13-s + (−24.9 − 43.2i)14-s + (12.6 − 21.9i)16-s + 63.6·17-s + (−57.9 − 1.63i)18-s + 5.41·19-s + ⋯
 L(s)  = 1 + (0.379 − 0.657i)2-s + (−0.512 + 0.858i)3-s + (0.212 + 0.367i)4-s + (0.369 + 0.662i)6-s + (0.627 − 1.08i)7-s + 1.08·8-s + (−0.475 − 0.879i)9-s + (−0.171 + 0.296i)11-s + (−0.424 − 0.00600i)12-s + (0.539 + 0.935i)13-s + (−0.476 − 0.825i)14-s + (0.197 − 0.342i)16-s + 0.908·17-s + (−0.758 − 0.0214i)18-s + 0.0653·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $0.948 - 0.315i$ 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: $$(2,\ 225,\ (\ :3/2),\ 0.948 - 0.315i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.22177 + 0.359435i$$ $$L(\frac12)$$ $$\approx$$ $$2.22177 + 0.359435i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (2.66 - 4.46i)T$$
5 $$1$$
good2 $$1 + (-1.07 + 1.85i)T + (-4 - 6.92i)T^{2}$$
7 $$1 + (-11.6 + 20.1i)T + (-171.5 - 297. i)T^{2}$$
11 $$1 + (6.24 - 10.8i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + (-25.3 - 43.8i)T + (-1.09e3 + 1.90e3i)T^{2}$$
17 $$1 - 63.6T + 4.91e3T^{2}$$
19 $$1 - 5.41T + 6.85e3T^{2}$$
23 $$1 + (-44.2 - 76.6i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + (72.2 - 125. i)T + (-1.21e4 - 2.11e4i)T^{2}$$
31 $$1 + (-138. - 240. i)T + (-1.48e4 + 2.57e4i)T^{2}$$
37 $$1 - 243.T + 5.06e4T^{2}$$
41 $$1 + (234. + 405. i)T + (-3.44e4 + 5.96e4i)T^{2}$$
43 $$1 + (-249. + 431. i)T + (-3.97e4 - 6.88e4i)T^{2}$$
47 $$1 + (226. - 391. i)T + (-5.19e4 - 8.99e4i)T^{2}$$
53 $$1 + 223.T + 1.48e5T^{2}$$
59 $$1 + (57.9 + 100. i)T + (-1.02e5 + 1.77e5i)T^{2}$$
61 $$1 + (41.3 - 71.7i)T + (-1.13e5 - 1.96e5i)T^{2}$$
67 $$1 + (405. + 701. i)T + (-1.50e5 + 2.60e5i)T^{2}$$
71 $$1 + 134.T + 3.57e5T^{2}$$
73 $$1 - 707.T + 3.89e5T^{2}$$
79 $$1 + (208. - 361. i)T + (-2.46e5 - 4.26e5i)T^{2}$$
83 $$1 + (-461. + 799. i)T + (-2.85e5 - 4.95e5i)T^{2}$$
89 $$1 - 114.T + 7.04e5T^{2}$$
97 $$1 + (180. - 313. 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

−11.71845620794396799107616360530, −10.87681774505045802157082696922, −10.39406869368736265429779767685, −9.145704028964125075086174797220, −7.78331337454630207320961112690, −6.78867293069390228764451743305, −5.15076262544433062799674635459, −4.20799584840607027205722080251, −3.34629005443114876443047209234, −1.37550585672371077827221600372, 1.07369170041312594241777947219, 2.54721445897063188406019456334, 4.85663961485999195239295522058, 5.77726748898889121291295667070, 6.28541847815552310916020221834, 7.75847351300572289674350087388, 8.234831817731694838886566225854, 9.922535271797209236269019246411, 11.10588421426745836167831072978, 11.68381719116397045391322512619