# Properties

 Label 2-15e2-15.2-c1-0-1 Degree $2$ Conductor $225$ Sign $-0.391 - 0.920i$ Analytic cond. $1.79663$ Root an. cond. $1.34038$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.707 + 0.707i)2-s + 0.999i·4-s + (2 + 2i)7-s + (−2.12 − 2.12i)8-s + 2.82i·11-s + (−1 + i)13-s − 2.82·14-s + 1.00·16-s + (−2.82 + 2.82i)17-s + (−2.00 − 2.00i)22-s + (2.82 + 2.82i)23-s − 1.41i·26-s + (−1.99 + 1.99i)28-s + 4.24·29-s − 4·31-s + (3.53 − 3.53i)32-s + ⋯
 L(s)  = 1 + (−0.499 + 0.499i)2-s + 0.499i·4-s + (0.755 + 0.755i)7-s + (−0.750 − 0.750i)8-s + 0.852i·11-s + (−0.277 + 0.277i)13-s − 0.755·14-s + 0.250·16-s + (−0.685 + 0.685i)17-s + (−0.426 − 0.426i)22-s + (0.589 + 0.589i)23-s − 0.277i·26-s + (−0.377 + 0.377i)28-s + 0.787·29-s − 0.718·31-s + (0.624 − 0.624i)32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $-0.391 - 0.920i$ Analytic conductor: $$1.79663$$ Root analytic conductor: $$1.34038$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{225} (107, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 225,\ (\ :1/2),\ -0.391 - 0.920i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.502559 + 0.759545i$$ $$L(\frac12)$$ $$\approx$$ $$0.502559 + 0.759545i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 + (0.707 - 0.707i)T - 2iT^{2}$$
7 $$1 + (-2 - 2i)T + 7iT^{2}$$
11 $$1 - 2.82iT - 11T^{2}$$
13 $$1 + (1 - i)T - 13iT^{2}$$
17 $$1 + (2.82 - 2.82i)T - 17iT^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + (-2.82 - 2.82i)T + 23iT^{2}$$
29 $$1 - 4.24T + 29T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 + (1 + i)T + 37iT^{2}$$
41 $$1 + 1.41iT - 41T^{2}$$
43 $$1 + (-8 + 8i)T - 43iT^{2}$$
47 $$1 + (-5.65 + 5.65i)T - 47iT^{2}$$
53 $$1 + (-2.82 - 2.82i)T + 53iT^{2}$$
59 $$1 - 8.48T + 59T^{2}$$
61 $$1 - 8T + 61T^{2}$$
67 $$1 + (4 + 4i)T + 67iT^{2}$$
71 $$1 + 5.65iT - 71T^{2}$$
73 $$1 + (1 - i)T - 73iT^{2}$$
79 $$1 + 12iT - 79T^{2}$$
83 $$1 + (-2.82 - 2.82i)T + 83iT^{2}$$
89 $$1 + 12.7T + 89T^{2}$$
97 $$1 + (-11 - 11i)T + 97iT^{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

−12.37564912344075466035146942045, −11.77595465101173436169065162725, −10.54996192158307526478698564559, −9.251053975480417444355696257082, −8.646272972011338520076968077525, −7.60199167602809397153443426663, −6.74815821322440349680070626144, −5.36123438529372113779581079301, −4.01941594741982611705807245109, −2.25848312992364474183984224563, 0.941842629029834829276222122057, 2.67939316892579796084197521046, 4.49273148215298350663438184775, 5.63791021885016944402708361816, 6.95251237299363432805196377715, 8.207357606127557863819765877889, 9.087039964230170588001187478423, 10.16410884146628574775607099986, 11.00891156892627751983405687857, 11.46959344371491948737042516898