# Properties

 Label 2-10e2-100.31-c0-0-0 Degree $2$ Conductor $100$ Sign $0.535 - 0.844i$ Analytic cond. $0.0499065$ Root an. cond. $0.223397$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)10-s + (−0.5 + 1.53i)13-s + (0.309 − 0.951i)16-s + (−0.5 − 0.363i)17-s + 0.999·18-s + 0.999·20-s + (0.309 + 0.951i)25-s − 1.61·26-s + (−0.5 + 0.363i)29-s + 32-s + ⋯
 L(s)  = 1 + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)10-s + (−0.5 + 1.53i)13-s + (0.309 − 0.951i)16-s + (−0.5 − 0.363i)17-s + 0.999·18-s + 0.999·20-s + (0.309 + 0.951i)25-s − 1.61·26-s + (−0.5 + 0.363i)29-s + 32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$100$$    =    $$2^{2} \cdot 5^{2}$$ Sign: $0.535 - 0.844i$ Analytic conductor: $$0.0499065$$ Root analytic conductor: $$0.223397$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{100} (31, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 100,\ (\ :0),\ 0.535 - 0.844i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5534260427$$ $$L(\frac12)$$ $$\approx$$ $$0.5534260427$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.309 - 0.951i)T$$
5 $$1 + (0.809 + 0.587i)T$$
good3 $$1 + (-0.309 + 0.951i)T^{2}$$
7 $$1 - T^{2}$$
11 $$1 + (0.809 - 0.587i)T^{2}$$
13 $$1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}$$
17 $$1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}$$
19 $$1 + (-0.309 - 0.951i)T^{2}$$
23 $$1 + (0.809 - 0.587i)T^{2}$$
29 $$1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2}$$
31 $$1 + (-0.309 - 0.951i)T^{2}$$
37 $$1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2}$$
41 $$1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2}$$
43 $$1 - T^{2}$$
47 $$1 + (-0.309 + 0.951i)T^{2}$$
53 $$1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}$$
59 $$1 + (0.809 + 0.587i)T^{2}$$
61 $$1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}$$
67 $$1 + (-0.309 - 0.951i)T^{2}$$
71 $$1 + (-0.309 + 0.951i)T^{2}$$
73 $$1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2}$$
79 $$1 + (-0.309 + 0.951i)T^{2}$$
83 $$1 + (-0.309 - 0.951i)T^{2}$$
89 $$1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}$$
97 $$1 + (0.5 - 0.363i)T + (0.309 - 0.951i)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

−14.53082763279916007951704825813, −13.33715435256026911803134895286, −12.30978471057418900351284262357, −11.56782324166309553129123243505, −9.499591714021149766685915326899, −8.795965928933043041405356203034, −7.44260981790956792459665870802, −6.52341597470470667371308755766, −4.84001626020136688398360006484, −3.82709231667603315271129119450, 2.66152548206980973020188319867, 4.14002826679925869875833378269, 5.52389601233870343120666487150, 7.40735181615435052329967335639, 8.524044957078315048199567134733, 10.21714847714142870445182399272, 10.72134589933074432226622561992, 11.84084510380340478198286773064, 12.83333147423805482282199149653, 13.73966495852089220025482417049