# Properties

 Label 2-4400-5.4-c1-0-29 Degree $2$ Conductor $4400$ Sign $0.447 - 0.894i$ Analytic cond. $35.1341$ Root an. cond. $5.92740$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2i·7-s + 3·9-s − 11-s + 4i·13-s − 4i·17-s + 6·29-s − 2i·37-s + 6·41-s + 2i·43-s + 3·49-s + 10i·53-s + 12·59-s − 6·61-s + 6i·63-s + 12i·67-s + ⋯
 L(s)  = 1 + 0.755i·7-s + 9-s − 0.301·11-s + 1.10i·13-s − 0.970i·17-s + 1.11·29-s − 0.328i·37-s + 0.937·41-s + 0.304i·43-s + 0.428·49-s + 1.37i·53-s + 1.56·59-s − 0.768·61-s + 0.755i·63-s + 1.46i·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{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: $$4400$$    =    $$2^{4} \cdot 5^{2} \cdot 11$$ Sign: $0.447 - 0.894i$ Analytic conductor: $$35.1341$$ Root analytic conductor: $$5.92740$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{4400} (4049, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 4400,\ (\ :1/2),\ 0.447 - 0.894i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.964527062$$ $$L(\frac12)$$ $$\approx$$ $$1.964527062$$ $$L(\frac{3}{2})$$ 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$$
11 $$1 + T$$
good3 $$1 - 3T^{2}$$
7 $$1 - 2iT - 7T^{2}$$
13 $$1 - 4iT - 13T^{2}$$
17 $$1 + 4iT - 17T^{2}$$
19 $$1 + 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 - 6T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + 2iT - 37T^{2}$$
41 $$1 - 6T + 41T^{2}$$
43 $$1 - 2iT - 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 - 10iT - 53T^{2}$$
59 $$1 - 12T + 59T^{2}$$
61 $$1 + 6T + 61T^{2}$$
67 $$1 - 12iT - 67T^{2}$$
71 $$1 + 16T + 71T^{2}$$
73 $$1 + 4iT - 73T^{2}$$
79 $$1 + 4T + 79T^{2}$$
83 $$1 - 2iT - 83T^{2}$$
89 $$1 + 6T + 89T^{2}$$
97 $$1 + 2iT - 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

−8.674672677514559886206668669138, −7.59511740168752741933802770536, −7.12343187604897774184741135825, −6.33400919780073255354975549937, −5.55342483593094509958958667837, −4.64513581055097513962666283690, −4.15604306452405247242730258228, −2.91709587576751559629954889162, −2.18669152782869049600699106230, −1.08153760318256068298168473569, 0.63704184630502587410224353937, 1.63037584494344685884396457514, 2.81483790011414725713286623105, 3.74587021815996209073081003393, 4.39947883964187868928118092063, 5.22389100571364498357948796597, 6.09681253561290229640267151864, 6.85045188555886514053510567617, 7.55839662007355384314449846099, 8.100250153413507128706688306754