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Properties

Label 1-4600-4600.1011-r0-0-0
Degree $1$
Conductor $4600$
Sign $0.876 - 0.481i$
Analytic cond. $21.3623$
Root an. cond. $21.3623$
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  + (−0.809 − 0.587i)3-s + 7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)33-s + (0.309 + 0.951i)37-s + (−0.309 + 0.951i)39-s + (0.309 + 0.951i)41-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + 7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)33-s + (0.309 + 0.951i)37-s + (−0.309 + 0.951i)39-s + (0.309 + 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(21.3623\)
Root analytic conductor: \(21.3623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4600,\ (0:\ ),\ 0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.587359178 - 0.4075645653i\)
\(L(\frac12)\) \(\approx\) \(1.587359178 - 0.4075645653i\)
\(L(1)\) \(\approx\) \(0.9963390063 - 0.1640116573i\)
\(L(1)\) \(\approx\) \(0.9963390063 - 0.1640116573i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−18.285169675303739639519117437699, −17.34408005845142384350173196837, −17.05577127581709659862502291292, −16.21610359816458862212865845716, −15.79637435231802054004247991715, −14.92747556543686852987473494629, −14.17597368658184880359371677262, −13.856663457696752081268501743060, −12.61765015940614818656424683574, −11.96732867702191504731158076678, −11.525709176371961019624785806817, −10.8140084742513129249226450727, −10.21736570178294973954830177487, −9.52147648836034123419557834356, −8.6283843650711450247802492888, −8.03785678417752470796748061266, −7.158581167838714054458484628930, −6.32394370372651264564434727737, −5.54211447506645503399015030832, −5.13901151535956772323755515712, −4.20566164685671479477313866480, −3.6737436400312749768599262736, −2.61830389950662045299727865698, −1.51425507054971130615111704803, −0.77078331001861965295540748112, 0.763507857430023898138635873313, 1.38026882451501608799832618891, 2.38039038364422929108222872365, 3.060041573389656433080263764827, 4.45413700717584093732917447955, 5.03147808557932291874506946911, 5.3753631369736677231551047737, 6.42956578998079462257874082420, 7.12898256704451679687359260691, 7.93786534706888520512242466392, 8.0500961183774898014075334983, 9.45358109715576804706610924172, 10.09873342892800011007065329357, 10.73705731459814147217488180972, 11.55948879558334936442688303813, 11.98340370526584601920618396022, 12.64882568198315089140097700933, 13.376851383046980821335821519349, 14.04485205090531689526504168860, 14.84258369154214498519040443222, 15.496332300296200332863107381989, 16.19895929998827376764005702727, 17.081077120722064809358761091686, 17.54747016166907517770657576596, 18.10473308570104906071371736419

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