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Properties

Label 1-5225-5225.831-r0-0-0
Degree $1$
Conductor $5225$
Sign $-0.100 + 0.994i$
Analytic cond. $24.2647$
Root an. cond. $24.2647$
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  + (−0.374 − 0.927i)2-s + (0.939 + 0.342i)3-s + (−0.719 + 0.694i)4-s + (−0.0348 − 0.999i)6-s + (−0.669 − 0.743i)7-s + (0.913 + 0.406i)8-s + (0.766 + 0.642i)9-s + (−0.913 + 0.406i)12-s + (−0.939 + 0.342i)13-s + (−0.438 + 0.898i)14-s + (0.0348 − 0.999i)16-s + (0.374 + 0.927i)17-s + (0.309 − 0.951i)18-s + (−0.374 − 0.927i)21-s + (0.438 + 0.898i)23-s + (0.719 + 0.694i)24-s + ⋯
L(s)  = 1  + (−0.374 − 0.927i)2-s + (0.939 + 0.342i)3-s + (−0.719 + 0.694i)4-s + (−0.0348 − 0.999i)6-s + (−0.669 − 0.743i)7-s + (0.913 + 0.406i)8-s + (0.766 + 0.642i)9-s + (−0.913 + 0.406i)12-s + (−0.939 + 0.342i)13-s + (−0.438 + 0.898i)14-s + (0.0348 − 0.999i)16-s + (0.374 + 0.927i)17-s + (0.309 − 0.951i)18-s + (−0.374 − 0.927i)21-s + (0.438 + 0.898i)23-s + (0.719 + 0.694i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $-0.100 + 0.994i$
Analytic conductor: \(24.2647\)
Root analytic conductor: \(24.2647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5225} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5225,\ (0:\ ),\ -0.100 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4912279430 + 0.5432570989i\)
\(L(\frac12)\) \(\approx\) \(0.4912279430 + 0.5432570989i\)
\(L(1)\) \(\approx\) \(0.8959151295 - 0.1614940562i\)
\(L(1)\) \(\approx\) \(0.8959151295 - 0.1614940562i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.374 - 0.927i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.374 + 0.927i)T \)
23 \( 1 + (0.438 + 0.898i)T \)
29 \( 1 + (-0.241 - 0.970i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.615 + 0.788i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (-0.990 + 0.139i)T \)
59 \( 1 + (0.241 - 0.970i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.374 - 0.927i)T \)
71 \( 1 + (0.719 + 0.694i)T \)
73 \( 1 + (-0.0348 + 0.999i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (-0.559 + 0.829i)T \)
97 \( 1 + (-0.848 - 0.529i)T \)
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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

−18.00295455677043202230693450526, −16.988980500433022293091993550578, −16.42758761631157394874516487355, −15.74041732698486002290261643283, −15.06138577022832659135929402236, −14.6621660458797018262943489813, −14.02256740682767074912084598175, −13.20873540397479310347406608339, −12.6768189456505675767419528588, −12.07201531780977292696850518086, −10.88851068814811843256504972763, −9.921963988989579886814373764762, −9.53788011106645967518289334823, −8.937160970674008975000387929602, −8.29428280023346294601647528930, −7.54680685023429470649321706444, −6.98318491765685773668686310466, −6.391652851685708827468368738056, −5.43703846312257350160124540321, −4.864859656258829918480547536223, −3.84702882567987325696496979156, −2.95601530428067982650021237969, −2.34165965590983191967501311069, −1.28502031948654022747606047639, −0.19618367951769783522291604619, 1.16415212139673775049800012853, 1.93945007355000372574230225483, 2.72190541787831762246382605045, 3.45868615835641744309535826860, 3.97615866733151975916435817606, 4.63146377634061608865864780978, 5.58853933343379282000858118137, 6.860004642725511767783287597742, 7.5150766666987802689687224314, 8.02775298952484425603584258699, 8.90881159838131895089679386550, 9.6293809157198542697287519337, 9.84423209398771970843190341651, 10.664707577246829967207166055242, 11.238328896894057600411088684651, 12.279469372875139867511567943053, 12.83438538785066283478234615067, 13.39322753706505392884395646446, 14.06629160447492582882795490231, 14.65216522652860561053525217904, 15.47247848167626920596306747898, 16.30567467784812301129427916682, 16.97156360817241355970584137940, 17.36723928305523043214488146845, 18.44252813101140379585069629913

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