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Properties

Label 1-5520-5520.2477-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.997 + 0.0650i$
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  + (−0.540 − 0.841i)7-s + (−0.989 − 0.142i)11-s + (0.841 + 0.540i)13-s + (−0.281 + 0.959i)17-s + (0.281 + 0.959i)19-s + (0.281 − 0.959i)29-s + (0.415 + 0.909i)31-s + (−0.654 − 0.755i)37-s + (−0.654 + 0.755i)41-s + (−0.415 + 0.909i)43-s − i·47-s + (−0.415 + 0.909i)49-s + (0.841 − 0.540i)53-s + (0.540 − 0.841i)59-s + (−0.909 + 0.415i)61-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)7-s + (−0.989 − 0.142i)11-s + (0.841 + 0.540i)13-s + (−0.281 + 0.959i)17-s + (0.281 + 0.959i)19-s + (0.281 − 0.959i)29-s + (0.415 + 0.909i)31-s + (−0.654 − 0.755i)37-s + (−0.654 + 0.755i)41-s + (−0.415 + 0.909i)43-s − i·47-s + (−0.415 + 0.909i)49-s + (0.841 − 0.540i)53-s + (0.540 − 0.841i)59-s + (−0.909 + 0.415i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.997 + 0.0650i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (2477, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.997 + 0.0650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001270001181 + 0.03900275792i\)
\(L(\frac12)\) \(\approx\) \(0.001270001181 + 0.03900275792i\)
\(L(1)\) \(\approx\) \(0.8261509487 + 0.01190171547i\)
\(L(1)\) \(\approx\) \(0.8261509487 + 0.01190171547i\)

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 \( 1 + (-0.540 - 0.841i)T \)
11 \( 1 + (-0.989 - 0.142i)T \)
13 \( 1 + (0.841 + 0.540i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (0.281 + 0.959i)T \)
29 \( 1 + (0.281 - 0.959i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (-0.654 - 0.755i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + (0.540 - 0.841i)T \)
61 \( 1 + (-0.909 + 0.415i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.281 - 0.959i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 + (-0.755 - 0.654i)T \)
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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

−17.689530638367963819403994917463, −16.86759342863633548610631060101, −15.95109859455780160241531961855, −15.6011794245028920104127883827, −15.24357002076559794152273754519, −14.147310978073427114231334842461, −13.38208948012879516213165453752, −13.090024210075831330335628834582, −12.19155903666850953908150076910, −11.64247694374933067407807108836, −10.815439176960541494592193746612, −10.22827278612551095615118422637, −9.42161192630411244256471047600, −8.78949992096985685422060849504, −8.22496686457825475273167117542, −7.27725710467309616592752772110, −6.71650727671967907899652440487, −5.78646278019163169366366576611, −5.28618617600721452471695443688, −4.58411401917896327899305874885, −3.45548708769566286101500955032, −2.83501228382849580520517816444, −2.28587622405096308739210892017, −1.09193273921313334637485107733, −0.01080846570291370657548159199, 1.189249816792236150232422123234, 1.93441108906551251467459764409, 3.01048720079070460292083359117, 3.67304931252562674516866020649, 4.27681010570204666083287528820, 5.18927099731530707089453393964, 6.04453858679823639404478282338, 6.56266861370040579172453368847, 7.35626170648312614121844187653, 8.166836944443262965712065728264, 8.5873993663804480742337493827, 9.64945812923568625974406885680, 10.26664201545103326339289600883, 10.68592630371096530831397504112, 11.49478234481712583617411829667, 12.285265509959351416941772902137, 13.04690440586273563592702488690, 13.52685112955194305603734523409, 14.06344660740020579081461557287, 14.95935930084593687778078870621, 15.63463266997375172928435608838, 16.38612438626226317625152657742, 16.611699760112265035653640551073, 17.63588025799211520502482059889, 18.104180081866597974520455829300

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