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Properties

Label 1-4928-4928.1579-r0-0-0
Degree $1$
Conductor $4928$
Sign $0.570 - 0.821i$
Analytic cond. $22.8855$
Root an. cond. $22.8855$
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  + (0.333 − 0.942i)3-s + (−0.0261 − 0.999i)5-s + (−0.777 − 0.629i)9-s + (0.852 + 0.522i)13-s + (−0.951 − 0.309i)15-s + (−0.994 − 0.104i)17-s + (0.182 + 0.983i)19-s + (0.258 + 0.965i)23-s + (−0.998 + 0.0523i)25-s + (−0.852 + 0.522i)27-s + (0.0784 + 0.996i)29-s + (0.913 + 0.406i)31-s + (−0.942 + 0.333i)37-s + (0.777 − 0.629i)39-s + (0.891 + 0.453i)41-s + ⋯
L(s)  = 1  + (0.333 − 0.942i)3-s + (−0.0261 − 0.999i)5-s + (−0.777 − 0.629i)9-s + (0.852 + 0.522i)13-s + (−0.951 − 0.309i)15-s + (−0.994 − 0.104i)17-s + (0.182 + 0.983i)19-s + (0.258 + 0.965i)23-s + (−0.998 + 0.0523i)25-s + (−0.852 + 0.522i)27-s + (0.0784 + 0.996i)29-s + (0.913 + 0.406i)31-s + (−0.942 + 0.333i)37-s + (0.777 − 0.629i)39-s + (0.891 + 0.453i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4928\)    =    \(2^{6} \cdot 7 \cdot 11\)
Sign: $0.570 - 0.821i$
Analytic conductor: \(22.8855\)
Root analytic conductor: \(22.8855\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4928} (1579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4928,\ (0:\ ),\ 0.570 - 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.635363068 - 0.8557912897i\)
\(L(\frac12)\) \(\approx\) \(1.635363068 - 0.8557912897i\)
\(L(1)\) \(\approx\) \(1.079797530 - 0.4418315727i\)
\(L(1)\) \(\approx\) \(1.079797530 - 0.4418315727i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.333 - 0.942i)T \)
5 \( 1 + (-0.0261 - 0.999i)T \)
13 \( 1 + (0.852 + 0.522i)T \)
17 \( 1 + (-0.994 - 0.104i)T \)
19 \( 1 + (0.182 + 0.983i)T \)
23 \( 1 + (0.258 + 0.965i)T \)
29 \( 1 + (0.0784 + 0.996i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.942 + 0.333i)T \)
41 \( 1 + (0.891 + 0.453i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (0.688 + 0.725i)T \)
59 \( 1 + (0.182 - 0.983i)T \)
61 \( 1 + (0.688 - 0.725i)T \)
67 \( 1 + (-0.793 - 0.608i)T \)
71 \( 1 + (0.156 + 0.987i)T \)
73 \( 1 + (0.544 - 0.838i)T \)
79 \( 1 + (0.994 - 0.104i)T \)
83 \( 1 + (-0.233 + 0.972i)T \)
89 \( 1 + (0.965 - 0.258i)T \)
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

−17.96687791786504791624803218952, −17.680478076216800965359137018166, −16.80600789206988658878485034405, −15.86343986588303280894775706377, −15.55434495357526795652288483742, −14.961100519593544302254539780577, −14.30935744679951365913183723710, −13.531594366569168032942172861221, −13.16037571604223970504956030340, −11.84052343254599883053766775222, −11.283259112006859283781268376185, −10.630356598486949355796762366898, −10.26682823684603878082679338587, −9.36571798320478306072122528897, −8.707949148661707916317442885549, −8.08136338882968822823680296107, −7.20273388944146875538356354198, −6.393674320161974393887382748512, −5.817232756514232211747738348253, −4.80262797639693379476896119247, −4.16576918108171815166131532270, −3.43908477365841175628150623525, −2.664058390510204692135602196650, −2.20007658207900169594984557424, −0.62563301002496390058413902906, 0.76222662733505295943318992030, 1.55754246304046692945804741231, 2.040119263284902631889169747217, 3.2631180334218390607892490553, 3.840924667039207120433191572522, 4.84794330943962174647787148893, 5.55425220739591595931078956572, 6.36542864362837309479249521177, 6.95474458358644302818189335122, 7.83748545087455658400919157883, 8.476709945592609547170220784436, 8.92389777554503410500715106512, 9.598754546039185117726191771416, 10.6247408576100970686629581802, 11.530713418560545445847192601806, 11.986754949306388941852842545145, 12.672131268306052557338766677397, 13.37956352003628405503047806225, 13.73039765941267205180481831039, 14.44968467808267965576620894273, 15.44410057100254585530270233651, 15.93738203393756428717163161371, 16.79406581786585223678280243463, 17.30112760359017003380804470285, 18.09777047033240672291292505649

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