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Properties

Label 1-469-469.156-r0-0-0
Degree $1$
Conductor $469$
Sign $0.908 - 0.417i$
Analytic cond. $2.17802$
Root an. cond. $2.17802$
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  + (−0.888 − 0.458i)2-s + (0.235 − 0.971i)3-s + (0.580 + 0.814i)4-s + (−0.327 + 0.945i)5-s + (−0.654 + 0.755i)6-s + (−0.142 − 0.989i)8-s + (−0.888 − 0.458i)9-s + (0.723 − 0.690i)10-s + (0.981 − 0.189i)11-s + (0.928 − 0.371i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (−0.327 + 0.945i)16-s + (−0.995 − 0.0950i)17-s + (0.580 + 0.814i)18-s + (0.0475 − 0.998i)19-s + ⋯
L(s)  = 1  + (−0.888 − 0.458i)2-s + (0.235 − 0.971i)3-s + (0.580 + 0.814i)4-s + (−0.327 + 0.945i)5-s + (−0.654 + 0.755i)6-s + (−0.142 − 0.989i)8-s + (−0.888 − 0.458i)9-s + (0.723 − 0.690i)10-s + (0.981 − 0.189i)11-s + (0.928 − 0.371i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (−0.327 + 0.945i)16-s + (−0.995 − 0.0950i)17-s + (0.580 + 0.814i)18-s + (0.0475 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.908 - 0.417i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.908 - 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8535420842 - 0.1865832086i\)
\(L(\frac12)\) \(\approx\) \(0.8535420842 - 0.1865832086i\)
\(L(1)\) \(\approx\) \(0.7367543450 - 0.1913273940i\)
\(L(1)\) \(\approx\) \(0.7367543450 - 0.1913273940i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.888 - 0.458i)T \)
3 \( 1 + (0.235 - 0.971i)T \)
5 \( 1 + (-0.327 + 0.945i)T \)
11 \( 1 + (0.981 - 0.189i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (-0.995 - 0.0950i)T \)
19 \( 1 + (0.0475 - 0.998i)T \)
23 \( 1 + (0.723 + 0.690i)T \)
29 \( 1 + T \)
31 \( 1 + (0.928 + 0.371i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.723 + 0.690i)T \)
53 \( 1 + (0.580 + 0.814i)T \)
59 \( 1 + (0.928 + 0.371i)T \)
61 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.327 - 0.945i)T \)
79 \( 1 + (0.928 - 0.371i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + T \)
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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

−24.26746217316617070733492926387, −23.02882731870903390176245305810, −22.357845382888152668225516288981, −20.9779984746252449491705518636, −20.382798916236137608289312388197, −19.76914613773269597234591791289, −18.958150501912402055124138795372, −17.41500839901338246933246920959, −17.15094329944490685585742503574, −16.14547886645545277457346555748, −15.50581840571251944794905673072, −14.78370176384692895691329418886, −13.7653799220137793617082975952, −12.33918629033125097755154178359, −11.41993285475065010139083108743, −10.36666089243908523567731696092, −9.67770692267474835509567692727, −8.54916189851885321805963262721, −8.41847821127489401037040925042, −6.96943439566148671462063965957, −5.74655187842528072169511184475, −4.85384393133628562673183538939, −3.840628460551919526485163912029, −2.34009964874968953168042507510, −0.81387544381326839339185757314, 1.03055859721572030918133432758, 2.26462371972865230941857524328, 3.02646107775377771744245400669, 4.21966137490132646063182309257, 6.417505989293624654932906229207, 6.810366097356400701398540635423, 7.63453712597954884023050033545, 8.79246080034335661542045431103, 9.36645798152170769119289089429, 10.81222467624903614285885155043, 11.50851669152618745632339269498, 12.06476082822229945266472048197, 13.31533220991475665833023785987, 14.13719856783940770532106641581, 15.16885845757898969152256761367, 16.206718094430190169201874829840, 17.5144810591842265832318328688, 17.74266913186033776611817273382, 18.88703413031499007030750595155, 19.49234381810316895404183710610, 19.783829588735245953626607269129, 21.19478914627807602075632509907, 22.006912602749894761711411447363, 22.92450362384298042056745569981, 23.99321854819504054267714503112

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