# Properties

 Degree $2$ Conductor $4100$ Sign $0.447 - 0.894i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.21i·3-s − 5.06i·7-s − 1.90·9-s − 2.55·11-s + 4.93i·13-s + 2.68i·17-s − 4.72·19-s − 11.2·21-s + 1.49i·23-s − 2.41i·27-s − 2.43·29-s + 3.19·31-s + 5.67i·33-s + 2.90i·37-s + 10.9·39-s + ⋯
 L(s)  = 1 − 1.27i·3-s − 1.91i·7-s − 0.636·9-s − 0.771·11-s + 1.36i·13-s + 0.651i·17-s − 1.08·19-s − 2.44·21-s + 0.311i·23-s − 0.465i·27-s − 0.451·29-s + 0.573·31-s + 0.987i·33-s + 0.478i·37-s + 1.75·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$4100$$    =    $$2^{2} \cdot 5^{2} \cdot 41$$ Sign: $0.447 - 0.894i$ Motivic weight: $$1$$ Character: $\chi_{4100} (1149, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4100,\ (\ :1/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.2805154092$$ $$L(\frac12)$$ $$\approx$$ $$0.2805154092$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
41 $$1 + T$$
good3 $$1 + 2.21iT - 3T^{2}$$
7 $$1 + 5.06iT - 7T^{2}$$
11 $$1 + 2.55T + 11T^{2}$$
13 $$1 - 4.93iT - 13T^{2}$$
17 $$1 - 2.68iT - 17T^{2}$$
19 $$1 + 4.72T + 19T^{2}$$
23 $$1 - 1.49iT - 23T^{2}$$
29 $$1 + 2.43T + 29T^{2}$$
31 $$1 - 3.19T + 31T^{2}$$
37 $$1 - 2.90iT - 37T^{2}$$
43 $$1 - 7.62iT - 43T^{2}$$
47 $$1 - 5.15iT - 47T^{2}$$
53 $$1 - 11.1iT - 53T^{2}$$
59 $$1 + 1.49T + 59T^{2}$$
61 $$1 - 1.06T + 61T^{2}$$
67 $$1 - 10.5iT - 67T^{2}$$
71 $$1 + 12.6T + 71T^{2}$$
73 $$1 + 8.28iT - 73T^{2}$$
79 $$1 - 4.28T + 79T^{2}$$
83 $$1 + 10.5iT - 83T^{2}$$
89 $$1 + 9.36T + 89T^{2}$$
97 $$1 + 3.37iT - 97T^{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

−8.185522916081055541125251910652, −7.73841724511835007547761677733, −7.07825926948405559634864023052, −6.60340729685096424960021235362, −5.93183669271804193892566885154, −4.44885601411351677678900631740, −4.24901197689745335932155434781, −2.98717680689894999762254306896, −1.82520777247152123741553858143, −1.19226306949221549833061345847, 0.079723207578526011100499935300, 2.16691143668347396865640440920, 2.82043650841908994701810052938, 3.61991917364469964722336342811, 4.72171079604294707217546263522, 5.34551333896264484511280943542, 5.67630851728877711506390553157, 6.68833894505254560158593753280, 7.85451807921042455915256289369, 8.524121729601726445899243735340