# Properties

 Label 2-3200-40.29-c1-0-39 Degree $2$ Conductor $3200$ Sign $0.894 + 0.447i$ 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 − 0.317·3-s − 2.89·9-s + 3.78i·11-s − 1.89i·17-s − 5.97i·19-s + 1.87·27-s − 1.20i·33-s − 6.79·41-s + 8.48·43-s + 7·49-s + 0.603i·51-s + 1.89i·57-s − 14.1i·59-s + 16.3·67-s + 15.6i·73-s + ⋯
 L(s)  = 1 − 0.183·3-s − 0.966·9-s + 1.14i·11-s − 0.460i·17-s − 1.37i·19-s + 0.360·27-s − 0.209i·33-s − 1.06·41-s + 1.29·43-s + 49-s + 0.0845i·51-s + 0.251i·57-s − 1.84i·59-s + 1.99·67-s + 1.83i·73-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3200$$    =    $$2^{7} \cdot 5^{2}$$ Sign: $0.894 + 0.447i$ Analytic conductor: $$25.5521$$ Root analytic conductor: $$5.05491$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{3200} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3200,\ (\ :1/2),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.350704721$$ $$L(\frac12)$$ $$\approx$$ $$1.350704721$$ $$L(\frac{3}{2})$$ 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 + 0.317T + 3T^{2}$$
7 $$1 - 7T^{2}$$
11 $$1 - 3.78iT - 11T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 + 1.89iT - 17T^{2}$$
19 $$1 + 5.97iT - 19T^{2}$$
23 $$1 - 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + 37T^{2}$$
41 $$1 + 6.79T + 41T^{2}$$
43 $$1 - 8.48T + 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 + 14.1iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 - 16.3T + 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 15.6iT - 73T^{2}$$
79 $$1 + 79T^{2}$$
83 $$1 - 17.0T + 83T^{2}$$
89 $$1 + 4.10T + 89T^{2}$$
97 $$1 - 10iT - 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

−8.668914596305290049484426446045, −7.85232747260750437667436887853, −7.02305980394499674493907932514, −6.49609101104016920933069268026, −5.39490057853500975458997152967, −4.92348751532406194567968311412, −3.95737574808788203148537093993, −2.85142967645754438315352807658, −2.11608160927965895114788012817, −0.58282748740622291405485056933, 0.822448319716665190293184541911, 2.16203483443736676637964150574, 3.23254197082269130964774556293, 3.86430474383328412304597001552, 5.03050800031715969172751699242, 5.92224054908356469652361546858, 6.11112508458919345055670173493, 7.29233525967086709467625753523, 8.143220732547945347364362844363, 8.601741694883686603043154626394