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 $0$

Origins

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Normalization:  

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

Graph of the $Z$-function along the critical line