# Properties

 Label 2-28e2-7.4-c1-0-1 Degree $2$ Conductor $784$ Sign $-0.266 - 0.963i$ Analytic cond. $6.26027$ Root an. cond. $2.50205$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1 + 1.73i)3-s + (−2 − 3.46i)5-s + (−0.499 − 0.866i)9-s + 7.99·15-s + (−1 + 1.73i)17-s + (1 + 1.73i)19-s + (4 + 6.92i)23-s + (−5.49 + 9.52i)25-s − 4.00·27-s + 2·29-s + (−2 + 3.46i)31-s + (3 + 5.19i)37-s + 2·41-s − 8·43-s + (−1.99 + 3.46i)45-s + ⋯
 L(s)  = 1 + (−0.577 + 0.999i)3-s + (−0.894 − 1.54i)5-s + (−0.166 − 0.288i)9-s + 2.06·15-s + (−0.242 + 0.420i)17-s + (0.229 + 0.397i)19-s + (0.834 + 1.44i)23-s + (−1.09 + 1.90i)25-s − 0.769·27-s + 0.371·29-s + (−0.359 + 0.622i)31-s + (0.493 + 0.854i)37-s + 0.312·41-s − 1.21·43-s + (−0.298 + 0.516i)45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$784$$    =    $$2^{4} \cdot 7^{2}$$ Sign: $-0.266 - 0.963i$ Analytic conductor: $$6.26027$$ Root analytic conductor: $$2.50205$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{784} (753, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 784,\ (\ :1/2),\ -0.266 - 0.963i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.433896 + 0.570350i$$ $$L(\frac12)$$ $$\approx$$ $$0.433896 + 0.570350i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
7 $$1$$
good3 $$1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2}$$
5 $$1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-5.5 - 9.52i)T^{2}$$
13 $$1 + 13T^{2}$$
17 $$1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 - 2T + 29T^{2}$$
31 $$1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 - 2T + 41T^{2}$$
43 $$1 + 8T + 43T^{2}$$
47 $$1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 6T + 83T^{2}$$
89 $$1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 - 2T + 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

−10.51626774879680361853368289425, −9.670811966317645681185918411209, −8.910297209910759403257150780537, −8.159118395749300509145893752973, −7.23279691262199210163764178735, −5.72014630663450837996309757767, −5.04246744718464189925255466788, −4.34731053421375344325537493103, −3.50346446316351625518264873692, −1.30355356123409053464692514177, 0.42843877870386109503346527148, 2.32847666281513009834910693752, 3.34226234510873272797157465887, 4.55496294889768908866852938258, 6.00757487158782994957037193588, 6.73584687160972200688912488859, 7.24167264687526580812488089890, 7.959165032623729722878548868736, 9.162424098480658488920603158947, 10.39634671333918661667202689782