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Properties

Label 1-469-469.145-r0-0-0
Degree $1$
Conductor $469$
Sign $0.382 - 0.923i$
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  + (−0.928 − 0.371i)2-s + (−0.327 − 0.945i)3-s + (0.723 + 0.690i)4-s + (−0.888 − 0.458i)5-s + (−0.0475 + 0.998i)6-s + (−0.415 − 0.909i)8-s + (−0.786 + 0.618i)9-s + (0.654 + 0.755i)10-s + (0.888 + 0.458i)11-s + (0.415 − 0.909i)12-s + (−0.995 + 0.0950i)13-s + (−0.142 + 0.989i)15-s + (0.0475 + 0.998i)16-s + (0.959 + 0.281i)17-s + (0.959 − 0.281i)18-s + (0.786 + 0.618i)19-s + ⋯
L(s)  = 1  + (−0.928 − 0.371i)2-s + (−0.327 − 0.945i)3-s + (0.723 + 0.690i)4-s + (−0.888 − 0.458i)5-s + (−0.0475 + 0.998i)6-s + (−0.415 − 0.909i)8-s + (−0.786 + 0.618i)9-s + (0.654 + 0.755i)10-s + (0.888 + 0.458i)11-s + (0.415 − 0.909i)12-s + (−0.995 + 0.0950i)13-s + (−0.142 + 0.989i)15-s + (0.0475 + 0.998i)16-s + (0.959 + 0.281i)17-s + (0.959 − 0.281i)18-s + (0.786 + 0.618i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5092891968 - 0.3403334927i\)
\(L(\frac12)\) \(\approx\) \(0.5092891968 - 0.3403334927i\)
\(L(1)\) \(\approx\) \(0.5273184542 - 0.2389707184i\)
\(L(1)\) \(\approx\) \(0.5273184542 - 0.2389707184i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.928 - 0.371i)T \)
3 \( 1 + (-0.327 - 0.945i)T \)
5 \( 1 + (-0.888 - 0.458i)T \)
11 \( 1 + (0.888 + 0.458i)T \)
13 \( 1 + (-0.995 + 0.0950i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (0.786 + 0.618i)T \)
23 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.580 - 0.814i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.723 - 0.690i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (0.654 - 0.755i)T \)
53 \( 1 + (-0.235 + 0.971i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (0.235 - 0.971i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (0.888 + 0.458i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−24.10008228247100368738865758969, −23.14811995981198624013276817585, −22.419321636916712778759360663899, −21.474451448588772798454807516726, −20.4015766711057232983322653652, −19.56637789713728314960083843439, −19.076379138917957152969869326727, −17.757671063890906381689276825130, −17.17498313521872728804624324700, −16.12495274088254073505632844259, −15.74151918189390166958820192618, −14.67934174582360315895755484998, −14.212618091060252476481642873366, −12.01955145187104942571575491676, −11.599597071933803345271264908393, −10.709977421835488716716704278020, −9.747128689422469823793745916680, −9.121576770569030122477577321554, −7.91754983329206230826334990912, −7.14557321069132349106700070223, −6.01330575050228230658154232301, −5.0317293013695305838032039908, −3.74734042497569072559466188648, −2.74954707434280554760274299882, −0.782603947971510513984328141167, 0.78861068504486432933360504012, 1.79053249812123426438157353364, 3.0796763880755442973026300830, 4.33234751745850715224861112580, 5.79347801925747668619772102747, 7.054935905577921897321741823531, 7.57490791385271487127630851190, 8.42809873414358321495201788532, 9.42569309828001427200002748891, 10.5194989606831922222025431419, 11.66677524066242662926377259263, 12.19331441276501988190680874522, 12.62485418956574223292664164180, 14.14757240395176314030711196164, 15.15831745172740407245880136955, 16.529612502318486974289219495679, 16.80610863633395469671427128437, 17.7626574307109460451624959812, 18.78011545695898061877122702258, 19.26527224483995853912789140471, 20.08232940701719808813910969049, 20.69138579216602711618100015523, 22.18100998479406436047612942272, 22.77498299461482851255125008081, 24.05157176391681208311939147035

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