# Properties

 Label 1-5520-5520.83-r0-0-0 Degree $1$ Conductor $5520$ Sign $-0.923 - 0.382i$ Analytic cond. $25.6347$ Root an. cond. $25.6347$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

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

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$5520$$    =    $$2^{4} \cdot 3 \cdot 5 \cdot 23$$ Sign: $-0.923 - 0.382i$ Analytic conductor: $$25.6347$$ Root analytic conductor: $$25.6347$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{5520} (83, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 5520,\ (0:\ ),\ -0.923 - 0.382i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.1247943109 - 0.6270378876i$$ $$L(\frac12)$$ $$\approx$$ $$0.1247943109 - 0.6270378876i$$ $$L(1)$$ $$\approx$$ $$0.9520448757 - 0.1428318337i$$ $$L(1)$$ $$\approx$$ $$0.9520448757 - 0.1428318337i$$

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

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

−18.466509286699373503823577224668, −17.28755007797030893245569719117, −16.98055527492897152208636620457, −16.17230877636824945958676316657, −15.45476511468727670315089469781, −14.92810254057316691224336571606, −14.149325040143510060560092555784, −13.64581013347014788309442040339, −12.84360505615858028455872578326, −12.09457876318469304737490385704, −11.52419742206885574853173998642, −10.84140367980433520614645331387, −10.29578400004789149077097858880, −9.27313268152440188029474592644, −8.68674827256718828558876047445, −8.09776295963341209382572967275, −7.51926145505361417939699503811, −6.392534625695015996145443177754, −5.85245876597449453827281136499, −5.27627159609354196203690072080, −4.323096391815153444186753201392, −3.66592568491082303660338761659, −2.77851610433243444191527176702, −1.9005708968671491793055495451, −1.29662577349750781685572822777, 0.15375936896457527692450269436, 1.3588446486231380430575698775, 1.902924238779031759400591056369, 3.04579208987535775589895118600, 3.63206884062746656749927324802, 4.67716926130855577233629661957, 5.04571158525139183103179744665, 5.886588219150380372548675957890, 6.88038476857581838453941570406, 7.39549616486510968619053058816, 8.13473939165974399689278635449, 8.64814231385407063285872159145, 9.66600117608773860229932095371, 10.343556101521723451664961405950, 10.79733225485399658754390229373, 11.51002054854977456265299620006, 12.37009096744975345634277460541, 12.93627462494060628613959786690, 13.60871261015954244135963765395, 14.31969688110548204594163063439, 14.968652539387391137057277031448, 15.4955008907157514483902149526, 16.32411102760014449853193715887, 16.97589794677834841331310467520, 17.65933799760513230649634460438