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Properties

Label 1-5520-5520.3563-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.645 + 0.763i$
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  + (0.989 − 0.142i)7-s + (−0.909 − 0.415i)11-s + (−0.142 + 0.989i)13-s + (−0.755 + 0.654i)17-s + (−0.755 − 0.654i)19-s + (−0.755 + 0.654i)29-s + (0.959 + 0.281i)31-s + (−0.841 + 0.540i)37-s + (0.841 + 0.540i)41-s + (0.959 − 0.281i)43-s − i·47-s + (0.959 − 0.281i)49-s + (0.142 + 0.989i)53-s + (0.989 + 0.142i)59-s + (0.281 − 0.959i)61-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)7-s + (−0.909 − 0.415i)11-s + (−0.142 + 0.989i)13-s + (−0.755 + 0.654i)17-s + (−0.755 − 0.654i)19-s + (−0.755 + 0.654i)29-s + (0.959 + 0.281i)31-s + (−0.841 + 0.540i)37-s + (0.841 + 0.540i)41-s + (0.959 − 0.281i)43-s − i·47-s + (0.959 − 0.281i)49-s + (0.142 + 0.989i)53-s + (0.989 + 0.142i)59-s + (0.281 − 0.959i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.645 + 0.763i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (3563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.645 + 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3336919525 + 0.7190251833i\)
\(L(\frac12)\) \(\approx\) \(0.3336919525 + 0.7190251833i\)
\(L(1)\) \(\approx\) \(0.9417666004 + 0.09552868701i\)
\(L(1)\) \(\approx\) \(0.9417666004 + 0.09552868701i\)

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

−17.61427041003848067414430302984, −17.27155305141190190679875575435, −16.14628414368233708838780054748, −15.624767451995116505135263564773, −15.00439846727663355738529484966, −14.43875296729183532651030259484, −13.64794697672310635523479077707, −12.93711148703940337193753429200, −12.417374716468316872869968505773, −11.55846738038288491336032548612, −10.9449668902475707536138445966, −10.35483508995016757731202273302, −9.66351421094258478396116673368, −8.7395721939845188817905327787, −8.11272671466331303168110516534, −7.59956675532743032952735667590, −6.88037023430008329888526854109, −5.74136545446816296098632401395, −5.40678019994184290835248360753, −4.49410205863885290474069566597, −3.97782551411447899829981035248, −2.63843557112985899460750677407, −2.37219469994181713808445991260, −1.32717197664327123125573142110, −0.20359752541745626715187760629, 1.104778109262005519758616410877, 2.02332684886135406822784039625, 2.53886008634048561014280336916, 3.66705316628917406055169378134, 4.445582393899794190633555986916, 4.9434650647249281420248838948, 5.7739347134314595423481897762, 6.62334847559617350977550972264, 7.248045360081716976252889278580, 8.09966184948244986954340900937, 8.620584351730101576571860808839, 9.23066441000510071581774694365, 10.26739259900820796631733751930, 10.89550682218737525089103665618, 11.25714294231613599834844473174, 12.12239968562817738631584993237, 12.83326479615111093206772164680, 13.6189049999937953985123842546, 14.024464094542278730579837940303, 14.93519836049357792623573415166, 15.34116306252452753691207440370, 16.14438472061630351573978200279, 16.88664067965791409762259596029, 17.44787500964811487754332939944, 18.04658906777424226598597806401

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