Properties

Label 1-5288-5288.4955-r0-0-0
Degree $1$
Conductor $5288$
Sign $0.857 + 0.514i$
Analytic cond. $24.5573$
Root an. cond. $24.5573$
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  + (0.959 − 0.281i)3-s + (−0.683 − 0.730i)5-s + (0.909 + 0.415i)7-s + (0.841 − 0.540i)9-s + (−0.327 + 0.945i)11-s + (−0.996 + 0.0855i)13-s + (−0.861 − 0.508i)15-s + (−0.749 − 0.662i)17-s + (−0.856 + 0.516i)19-s + (0.989 + 0.142i)21-s + (0.917 − 0.398i)23-s + (−0.0665 + 0.997i)25-s + (0.654 − 0.755i)27-s + (−0.888 − 0.458i)29-s + (0.516 + 0.856i)31-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)3-s + (−0.683 − 0.730i)5-s + (0.909 + 0.415i)7-s + (0.841 − 0.540i)9-s + (−0.327 + 0.945i)11-s + (−0.996 + 0.0855i)13-s + (−0.861 − 0.508i)15-s + (−0.749 − 0.662i)17-s + (−0.856 + 0.516i)19-s + (0.989 + 0.142i)21-s + (0.917 − 0.398i)23-s + (−0.0665 + 0.997i)25-s + (0.654 − 0.755i)27-s + (−0.888 − 0.458i)29-s + (0.516 + 0.856i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5288 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5288\)    =    \(2^{3} \cdot 661\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(24.5573\)
Root analytic conductor: \(24.5573\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5288} (4955, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5288,\ (0:\ ),\ 0.857 + 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.917491414 + 0.5310902007i\)
\(L(\frac12)\) \(\approx\) \(1.917491414 + 0.5310902007i\)
\(L(1)\) \(\approx\) \(1.295628114 - 0.06774654297i\)
\(L(1)\) \(\approx\) \(1.295628114 - 0.06774654297i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
661 \( 1 \)
good3 \( 1 + (0.959 - 0.281i)T \)
5 \( 1 + (-0.683 - 0.730i)T \)
7 \( 1 + (0.909 + 0.415i)T \)
11 \( 1 + (-0.327 + 0.945i)T \)
13 \( 1 + (-0.996 + 0.0855i)T \)
17 \( 1 + (-0.749 - 0.662i)T \)
19 \( 1 + (-0.856 + 0.516i)T \)
23 \( 1 + (0.917 - 0.398i)T \)
29 \( 1 + (-0.888 - 0.458i)T \)
31 \( 1 + (0.516 + 0.856i)T \)
37 \( 1 + (0.508 + 0.861i)T \)
41 \( 1 + (-0.830 + 0.556i)T \)
43 \( 1 + (-0.640 - 0.768i)T \)
47 \( 1 + (0.483 - 0.875i)T \)
53 \( 1 + (0.345 + 0.938i)T \)
59 \( 1 + (-0.0855 + 0.996i)T \)
61 \( 1 + (0.743 - 0.669i)T \)
67 \( 1 + (0.170 + 0.985i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (-0.290 - 0.956i)T \)
79 \( 1 + (-0.836 + 0.548i)T \)
83 \( 1 + (-0.226 + 0.974i)T \)
89 \( 1 + (0.814 - 0.580i)T \)
97 \( 1 + (0.797 - 0.603i)T \)
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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

−17.930836291743853596243787882258, −17.2166244138875023777294576904, −16.55277152703750998855966773234, −15.5953354801482673133961506319, −15.16714287805956804296831567354, −14.61069925272786095070759262700, −14.171624392163854866350160191153, −13.227701185215244937176980151469, −12.86247056612279324616341004019, −11.62792141875909104021021222700, −11.0640463492403684354136415386, −10.65154088085837130720968706226, −9.86804508382759825645535036996, −8.92487580887020928637278201334, −8.36941346664470905012293702101, −7.74575998621622029101143021331, −7.22185207512776644229753291490, −6.47718387790170220746719271835, −5.3051330019967367177058240527, −4.55396207868700432569267203673, −3.955442278156405593945449746958, −3.23235746990167572653667098005, −2.459967558922951790146895421644, −1.8452368346707580270090503634, −0.495263499525969649727079613706, 0.91397883182427819669735375758, 1.9330617165057317347970027389, 2.3268214223554687336767785480, 3.28901391240532287946839289117, 4.39235575670660827021447301561, 4.630576656433729446467848212670, 5.362374898302803572655253001866, 6.71086974263712228886372209805, 7.287019666674701935543839363356, 7.87743610368942311810017025696, 8.575603419886874190484704864463, 8.970091653719789246939531596669, 9.804518344832963361075432130936, 10.52347853748813851705586122109, 11.640835216785672493964830855305, 12.00561037304352922626627686083, 12.760663421172116849383006212460, 13.22105484849812345560135914477, 14.11885972240598464563411204527, 14.92566656342026748003035275607, 15.16813052837600637886823671272, 15.70807884972293373838713636436, 16.89719023541671754501825690282, 17.18826639649854096337322631238, 18.26078859423637897539875230167

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