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Properties

Label 1-2420-2420.307-r1-0-0
Degree $1$
Conductor $2420$
Sign $0.434 - 0.900i$
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  − i·3-s + (0.755 − 0.654i)7-s − 9-s + (−0.909 + 0.415i)13-s + (0.281 + 0.959i)17-s + (0.959 + 0.281i)19-s + (0.654 + 0.755i)21-s + (−0.755 − 0.654i)23-s − i·27-s + (−0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.909 + 0.415i)37-s + (−0.415 − 0.909i)39-s + (−0.841 − 0.540i)41-s + (0.989 + 0.142i)43-s + ⋯
L(s)  = 1  − i·3-s + (0.755 − 0.654i)7-s − 9-s + (−0.909 + 0.415i)13-s + (0.281 + 0.959i)17-s + (0.959 + 0.281i)19-s + (0.654 + 0.755i)21-s + (−0.755 − 0.654i)23-s − i·27-s + (−0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (0.909 + 0.415i)37-s + (−0.415 − 0.909i)39-s + (−0.841 − 0.540i)41-s + (0.989 + 0.142i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2420\)    =    \(2^{2} \cdot 5 \cdot 11^{2}\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(260.065\)
Root analytic conductor: \(260.065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2420} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2420,\ (1:\ ),\ 0.434 - 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7856740961 - 0.4931671948i\)
\(L(\frac12)\) \(\approx\) \(0.7856740961 - 0.4931671948i\)
\(L(1)\) \(\approx\) \(0.9353917569 + 0.2433355859i\)
\(L(1)\) \(\approx\) \(0.9353917569 + 0.2433355859i\)

Euler product

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

−19.34629773392931338165045179148, −18.684756124939341454299984869856, −17.98750896832087083761705104403, −17.66029899230213435909622246310, −16.74932042330852792369498269539, −15.92489390992988594667598233236, −14.970924692631640466208086755331, −14.427850784857751681495387913729, −13.74343025975560734337353017113, −12.91761671625200805662129180453, −12.24861623485232931452409653007, −11.515041139648318414006992780954, −11.187506261961813259431706513374, −9.7594047829285867884037761199, −9.272050806205278112567865199982, −8.22091141331880553313281979272, −7.59824278080884580591621627926, −7.1754590807052419209136084019, −5.92101385894650922463625121576, −5.45870975397690298046921588979, −4.66065149289636305964314421210, −3.30576493951977445083930519149, −2.52489556304668370836122691638, −1.811301984893673190941970344443, −0.836350055007679689241663868363, 0.1750695111519797593324038514, 1.44890234496855593799183517783, 2.40542693450707102522702057814, 3.53594930964023523867864057641, 4.11715193108377062229073489903, 4.9181838400840152831707298626, 5.5454391905375584970057648, 6.55079905607795275196853253347, 7.597904951363179808922844340214, 8.15095130249099109206811886167, 9.07214987198430569478586845219, 9.86845357241737867907827026650, 10.39635441418596881186536056633, 11.14945906775322157092973597941, 11.83153609579255662937463405367, 12.608899020615746398102007210029, 13.78222983806237062687430928838, 14.32930664803733844933724422709, 14.83752589130373332033870003879, 15.611952757812184327145135360470, 16.59277219713023003564285704469, 16.84947744258576681056833526225, 17.644739554766863513262838507553, 18.4136168737436230038042444969, 19.43137355592751888247445852397

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