# Properties

 Label 2-648-648.115-c0-0-0 Degree $2$ Conductor $648$ Sign $-0.135 + 0.990i$ Analytic cond. $0.323394$ Root an. cond. $0.568677$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.396 − 0.918i)2-s + (0.893 − 0.448i)3-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)6-s + (−0.939 + 0.342i)8-s + (0.597 − 0.802i)9-s + (1.14 + 0.754i)11-s + (−0.939 − 0.342i)12-s + (−0.0581 + 0.998i)16-s + (−1.52 + 1.27i)17-s + (−0.500 − 0.866i)18-s + (−1.28 − 1.07i)19-s + (1.14 − 0.754i)22-s + (−0.686 + 0.727i)24-s + (−0.993 + 0.116i)25-s + ⋯
 L(s)  = 1 + (0.396 − 0.918i)2-s + (0.893 − 0.448i)3-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)6-s + (−0.939 + 0.342i)8-s + (0.597 − 0.802i)9-s + (1.14 + 0.754i)11-s + (−0.939 − 0.342i)12-s + (−0.0581 + 0.998i)16-s + (−1.52 + 1.27i)17-s + (−0.500 − 0.866i)18-s + (−1.28 − 1.07i)19-s + (1.14 − 0.754i)22-s + (−0.686 + 0.727i)24-s + (−0.993 + 0.116i)25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$648$$    =    $$2^{3} \cdot 3^{4}$$ Sign: $-0.135 + 0.990i$ Analytic conductor: $$0.323394$$ Root analytic conductor: $$0.568677$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{648} (115, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 648,\ (\ :0),\ -0.135 + 0.990i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.358444698$$ $$L(\frac12)$$ $$\approx$$ $$1.358444698$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.396 + 0.918i)T$$
3 $$1 + (-0.893 + 0.448i)T$$
good5 $$1 + (0.993 - 0.116i)T^{2}$$
7 $$1 + (-0.893 - 0.448i)T^{2}$$
11 $$1 + (-1.14 - 0.754i)T + (0.396 + 0.918i)T^{2}$$
13 $$1 + (-0.973 - 0.230i)T^{2}$$
17 $$1 + (1.52 - 1.27i)T + (0.173 - 0.984i)T^{2}$$
19 $$1 + (1.28 + 1.07i)T + (0.173 + 0.984i)T^{2}$$
23 $$1 + (-0.893 + 0.448i)T^{2}$$
29 $$1 + (0.286 - 0.957i)T^{2}$$
31 $$1 + (0.835 + 0.549i)T^{2}$$
37 $$1 + (0.939 - 0.342i)T^{2}$$
41 $$1 + (-0.707 - 1.64i)T + (-0.686 + 0.727i)T^{2}$$
43 $$1 + (-0.707 + 0.355i)T + (0.597 - 0.802i)T^{2}$$
47 $$1 + (0.835 - 0.549i)T^{2}$$
53 $$1 + (0.5 - 0.866i)T^{2}$$
59 $$1 + (-0.0971 + 0.0639i)T + (0.396 - 0.918i)T^{2}$$
61 $$1 + (0.0581 + 0.998i)T^{2}$$
67 $$1 + (-1.16 + 1.56i)T + (-0.286 - 0.957i)T^{2}$$
71 $$1 + (-0.766 - 0.642i)T^{2}$$
73 $$1 + (1.12 - 0.408i)T + (0.766 - 0.642i)T^{2}$$
79 $$1 + (0.686 + 0.727i)T^{2}$$
83 $$1 + (-0.137 + 0.318i)T + (-0.686 - 0.727i)T^{2}$$
89 $$1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2}$$
97 $$1 + (-0.0333 + 0.572i)T + (-0.993 - 0.116i)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

−10.63116849617570988261757878145, −9.531090666022575825633733501488, −9.017472445890334369896893697320, −8.206066929899026227477642147435, −6.81923203993889202170599312770, −6.17880278888877936853554286166, −4.39069880315795297957636059758, −4.00511425290022442036443309756, −2.52149860568947923448255700738, −1.67990703480025224177193494747, 2.43172516855255334981062552482, 3.84330283078782636166832966621, 4.29391630568579765366176441828, 5.62528201469657750610477159018, 6.61719362663302468043039272866, 7.46647380153620511062042786542, 8.553845213891898032951248627334, 8.927902064064333592374109322316, 9.789076582628220501039432000450, 10.97793103197784265593241307545