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Properties

Label 1-4760-4760.3467-r0-0-0
Degree $1$
Conductor $4760$
Sign $0.0932 - 0.995i$
Analytic cond. $22.1053$
Root an. cond. $22.1053$
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  + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s − i·13-s + (0.5 + 0.866i)19-s + (−0.866 + 0.5i)23-s − i·27-s − 29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)33-s + (0.866 − 0.5i)37-s + (0.5 − 0.866i)39-s − 41-s − i·43-s + (−0.866 + 0.5i)47-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s − i·13-s + (0.5 + 0.866i)19-s + (−0.866 + 0.5i)23-s − i·27-s − 29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)33-s + (0.866 − 0.5i)37-s + (0.5 − 0.866i)39-s − 41-s − i·43-s + (−0.866 + 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0932 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0932 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4760\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.0932 - 0.995i$
Analytic conductor: \(22.1053\)
Root analytic conductor: \(22.1053\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4760} (3467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4760,\ (0:\ ),\ 0.0932 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6374561723 - 0.5805633136i\)
\(L(\frac12)\) \(\approx\) \(0.6374561723 - 0.5805633136i\)
\(L(1)\) \(\approx\) \(0.7555495147 - 0.1224501572i\)
\(L(1)\) \(\approx\) \(0.7555495147 - 0.1224501572i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 - iT \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 - T \)
43 \( 1 - iT \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 - iT \)
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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

−18.07186360446231156740849192806, −17.683985405200171294277969456235, −16.93731656340374479520403671493, −16.39626363029618915897588527346, −15.637006412464374936798116668800, −14.98054153377221538665130327473, −14.63861604686894547809165913938, −13.36103905882637459422593216259, −12.93039130552151634283131561327, −12.07047338462559938751647929285, −11.5865440393045223390113744793, −10.88409875321197814385383210999, −10.10708118064771951567745593182, −9.650046194717866273055988556332, −8.94950599505431801626009181707, −7.88912040622699313713411523854, −7.2515348809440722197950575704, −6.42569181939965643706750656424, −5.814613696134315668800640373874, −5.02989656156144507578150947856, −4.424005607478592218447283338322, −3.6811937051441199979295492387, −2.78900261918276340834455276074, −1.72596763021167934350696960600, −0.77354165778153349710655747949, 0.34371376982858130822018947481, 1.58990149882787491776402906218, 1.86390333217813859699632900761, 3.293752985867684800158421637527, 3.92040438541194395013475846527, 4.85848571210931725152837511523, 5.615768621562762531128160190926, 6.2005727261773014236227388993, 6.83140814657671626372579825330, 7.61855642907967870990665440480, 8.264986030078247219671464907824, 9.20475385004341429288697126029, 9.84109563872722894896349648419, 10.73205256643556337239306061244, 11.37399148374708541138220614400, 11.81806539476016325591223842681, 12.49640183601520885142461324181, 13.24893815564094624201647647572, 14.03116404714046142956916914148, 14.34931339680860715349512891062, 15.50674819076666153490388421759, 16.32318584949015610262596908278, 16.553278563664261449770568991874, 17.26249797103574296165270269735, 18.072901253023409251549230598260

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