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Properties

Label 1-2420-2420.1299-r1-0-0
Degree $1$
Conductor $2420$
Sign $-0.943 + 0.331i$
Analytic cond. $260.065$
Root an. cond. $260.065$
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  + 3-s + (0.415 − 0.909i)7-s + 9-s + (−0.841 − 0.540i)13-s + (0.142 + 0.989i)17-s + (0.142 − 0.989i)19-s + (0.415 − 0.909i)21-s + (0.415 + 0.909i)23-s + 27-s + (−0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.841 + 0.540i)37-s + (−0.841 − 0.540i)39-s + (−0.959 − 0.281i)41-s + (−0.654 − 0.755i)43-s + ⋯
L(s)  = 1  + 3-s + (0.415 − 0.909i)7-s + 9-s + (−0.841 − 0.540i)13-s + (0.142 + 0.989i)17-s + (0.142 − 0.989i)19-s + (0.415 − 0.909i)21-s + (0.415 + 0.909i)23-s + 27-s + (−0.142 + 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.841 + 0.540i)37-s + (−0.841 − 0.540i)39-s + (−0.959 − 0.281i)41-s + (−0.654 − 0.755i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2420\)    =    \(2^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-0.943 + 0.331i$
Analytic conductor: \(260.065\)
Root analytic conductor: \(260.065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2420} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2420,\ (1:\ ),\ -0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.006105587045 - 0.03583429751i\)
\(L(\frac12)\) \(\approx\) \(0.006105587045 - 0.03583429751i\)
\(L(1)\) \(\approx\) \(1.284867945 - 0.1276392401i\)
\(L(1)\) \(\approx\) \(1.284867945 - 0.1276392401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + T \)
7 \( 1 + (0.415 - 0.909i)T \)
13 \( 1 + (-0.841 - 0.540i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (-0.142 + 0.989i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (-0.841 + 0.540i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (-0.959 + 0.281i)T \)
53 \( 1 + (-0.415 + 0.909i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.654 + 0.755i)T \)
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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

−19.66738382238884892251353280689, −18.9900183656170483883687982847, −18.49750460283346340298538543892, −17.85086088458480091658973869285, −16.69835405581226869231776182443, −16.168414046205258792358767237719, −15.20431144943540726802372366752, −14.71204710349239783543388416652, −14.19242692041888278615648733598, −13.35492989480385399286822954656, −12.50327465622886182700194848471, −11.92624161571022914317965331077, −11.12420583342777524823794576810, −9.88297986686782826281461938873, −9.59827618426761162153043153358, −8.6646426491144565889464703620, −8.11898684293422624748651071131, −7.30691876290772269093956037170, −6.554480846697599530033132010813, −5.40694714423781588139470645299, −4.740388387647822517260651187716, −3.82757412581193939050830287001, −2.86092810501741499515021562414, −2.21999028325802496863171523786, −1.46808158966459679492744664439, 0.004440455683937832151485657980, 1.30832828606616584726074747929, 1.90333839021672694827414724398, 3.17928411465369374894053727811, 3.536347031621143511091971425574, 4.68408896426315054384675196844, 5.18507047276219849405421927147, 6.59650075966746180540236987275, 7.318468498520154007878902189805, 7.78055514247767146453384349607, 8.69922215913939248789432796923, 9.31201754030660999189184035545, 10.3543708167397283600230193605, 10.604868076167926287659106069676, 11.765738704070037673521119824171, 12.65830516729744227789953697543, 13.33732230914257093370707407164, 13.86973004091458307794178498760, 14.80779014429707551622470434410, 15.062561846694834793822790520059, 15.99975536502727122906393671742, 16.91497051295207337323351890971, 17.52148222813300941616399578543, 18.22701585778491647028444163485, 19.24868054687182595144230532285

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