# Properties

 Label 2-3200-8.5-c1-0-75 Degree $2$ Conductor $3200$ Sign $-i$ Analytic cond. $25.5521$ Root an. cond. $5.05491$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.14i·3-s − 6.89·9-s − 6.61i·11-s − 7.89·17-s + 2.51i·19-s + 12.2i·27-s − 20.7·33-s + 12.7·41-s − 8.48i·43-s − 7·49-s + 24.8i·51-s + 7.89·57-s + 14.1i·59-s − 7.88i·67-s − 13.6·73-s + ⋯
 L(s)  = 1 − 1.81i·3-s − 2.29·9-s − 1.99i·11-s − 1.91·17-s + 0.575i·19-s + 2.36i·27-s − 3.62·33-s + 1.99·41-s − 1.29i·43-s − 49-s + 3.48i·51-s + 1.04·57-s + 1.84i·59-s − 0.962i·67-s − 1.60·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3200$$    =    $$2^{7} \cdot 5^{2}$$ Sign: $-i$ Analytic conductor: $$25.5521$$ Root analytic conductor: $$5.05491$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3200} (1601, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3200,\ (\ :1/2),\ -i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5097605613$$ $$L(\frac12)$$ $$\approx$$ $$0.5097605613$$ $$L(\frac{3}{2})$$ 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$$
good3 $$1 + 3.14iT - 3T^{2}$$
7 $$1 + 7T^{2}$$
11 $$1 + 6.61iT - 11T^{2}$$
13 $$1 - 13T^{2}$$
17 $$1 + 7.89T + 17T^{2}$$
19 $$1 - 2.51iT - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 - 37T^{2}$$
41 $$1 - 12.7T + 41T^{2}$$
43 $$1 + 8.48iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 - 14.1iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 + 7.88iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 13.6T + 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 14.1iT - 83T^{2}$$
89 $$1 - 13.8T + 89T^{2}$$
97 $$1 + 10T + 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

−8.042511493470553629528071090085, −7.38224003673974474936706578105, −6.50856756334546550174555864997, −6.11632274523036361135323372487, −5.41494446711332131873101037204, −4.04611958600484910536531856053, −2.96168533417526199989981647943, −2.23603266924591935745587948256, −1.13753538435333575524224412542, −0.16168061828592726358013354107, 2.08213856787426906996728661832, 2.91413828786457210999861216894, 4.13419129508342722091459242759, 4.52728873781391495617902336603, 4.99921577116159331332355433200, 6.12260008916806473155457670815, 6.92164348675091749525543909020, 7.84830134758384981043218566044, 8.845778431296596599617255638951, 9.351897823004443386940357212745