# Properties

 Label 1-5520-5520.77-r0-0-0 Degree $1$ Conductor $5520$ Sign $-0.554 - 0.832i$ Analytic cond. $25.6347$ Root an. cond. $25.6347$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.909 − 0.415i)7-s + (−0.281 − 0.959i)11-s + (0.415 − 0.909i)13-s + (−0.540 − 0.841i)17-s + (0.540 − 0.841i)19-s + (0.540 + 0.841i)29-s + (−0.654 − 0.755i)31-s + (−0.142 − 0.989i)37-s + (−0.142 + 0.989i)41-s + (0.654 − 0.755i)43-s − i·47-s + (0.654 − 0.755i)49-s + (0.415 + 0.909i)53-s + (−0.909 − 0.415i)59-s + (0.755 − 0.654i)61-s + ⋯
 L(s)  = 1 + (0.909 − 0.415i)7-s + (−0.281 − 0.959i)11-s + (0.415 − 0.909i)13-s + (−0.540 − 0.841i)17-s + (0.540 − 0.841i)19-s + (0.540 + 0.841i)29-s + (−0.654 − 0.755i)31-s + (−0.142 − 0.989i)37-s + (−0.142 + 0.989i)41-s + (0.654 − 0.755i)43-s − i·47-s + (0.654 − 0.755i)49-s + (0.415 + 0.909i)53-s + (−0.909 − 0.415i)59-s + (0.755 − 0.654i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$5520$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 23$$ Sign: $-0.554 - 0.832i$ Analytic conductor: $$25.6347$$ Root analytic conductor: $$25.6347$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{5520} (77, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 5520,\ (0:\ ),\ -0.554 - 0.832i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8323004867 - 1.554689152i$$ $$L(\frac12)$$ $$\approx$$ $$0.8323004867 - 1.554689152i$$ $$L(1)$$ $$\approx$$ $$1.091611007 - 0.3791756079i$$ $$L(1)$$ $$\approx$$ $$1.091611007 - 0.3791756079i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
23 $$1$$
good7 $$1 + (0.909 - 0.415i)T$$
11 $$1 + (-0.281 - 0.959i)T$$
13 $$1 + (0.415 - 0.909i)T$$
17 $$1 + (-0.540 - 0.841i)T$$
19 $$1 + (0.540 - 0.841i)T$$
29 $$1 + (0.540 + 0.841i)T$$
31 $$1 + (-0.654 - 0.755i)T$$
37 $$1 + (-0.142 - 0.989i)T$$
41 $$1 + (-0.142 + 0.989i)T$$
43 $$1 + (0.654 - 0.755i)T$$
47 $$1 - iT$$
53 $$1 + (0.415 + 0.909i)T$$
59 $$1 + (-0.909 - 0.415i)T$$
61 $$1 + (0.755 - 0.654i)T$$
67 $$1 + (0.959 + 0.281i)T$$
71 $$1 + (-0.959 - 0.281i)T$$
73 $$1 + (-0.540 + 0.841i)T$$
79 $$1 + (-0.415 + 0.909i)T$$
83 $$1 + (0.142 + 0.989i)T$$
89 $$1 + (0.654 - 0.755i)T$$
97 $$1 + (-0.989 - 0.142i)T$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−17.87921375531387285515330660002, −17.66502086193409655976220276517, −16.84005261055216302967655939049, −16.011975236461480437203344405862, −15.483156505288377025226058817988, −14.70229552769186705277459345799, −14.29988273234285594934038962689, −13.49767347774308703120364638792, −12.7411144948724116477182915835, −12.00325932146131649822960949932, −11.59270688004640898088989278193, −10.71004381836395992147229985865, −10.17087448225796691245708051020, −9.27298391222365683532630024807, −8.68528597826791586527312679306, −7.96244645305264808892915000660, −7.359935030540501118820903982950, −6.47941633134664692778008469172, −5.82916017745268369896579270423, −4.94588213313595813726180272251, −4.39645312119416837061169454841, −3.672837110367262958594596732697, −2.562146734383261514793763947097, −1.82399044809312324134171924593, −1.31374636608190535517338433078, 0.47544799159750897931102007756, 1.13296735133107811792761627684, 2.24582822110366401788209188950, 2.99049715389395695413471643345, 3.74856847041665824508404786721, 4.63229177481780170027575651453, 5.32580715249099795199885489638, 5.83785457937755370219128836497, 6.95691443647881159809244795382, 7.43074152908315967640499229921, 8.28085841807807897152606025627, 8.73678974005796221641790340874, 9.56137888429966193630949533127, 10.47229293767520068838337445975, 11.11074904696965140630117579597, 11.35062704447144005650679244746, 12.34394252006682280113357973210, 13.169767785935456399218578005468, 13.69984473708027230946625842431, 14.21880276909690593696770171350, 15.04110706082976937705263434629, 15.733369145782409596594248027490, 16.230471257059195883889403583201, 17.06267107223975431835407901413, 17.7149568815602519312525117208