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Properties

Label 1-469-469.229-r0-0-0
Degree $1$
Conductor $469$
Sign $0.663 - 0.747i$
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  + (0.959 + 0.281i)2-s + (−0.786 − 0.618i)3-s + (0.841 + 0.540i)4-s + (0.580 − 0.814i)5-s + (−0.580 − 0.814i)6-s + (0.654 + 0.755i)8-s + (0.235 + 0.971i)9-s + (0.786 − 0.618i)10-s + (−0.415 − 0.909i)11-s + (−0.327 − 0.945i)12-s + (−0.327 − 0.945i)13-s + (−0.959 + 0.281i)15-s + (0.415 + 0.909i)16-s + (0.888 + 0.458i)17-s + (−0.0475 + 0.998i)18-s + (0.959 − 0.281i)19-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)2-s + (−0.786 − 0.618i)3-s + (0.841 + 0.540i)4-s + (0.580 − 0.814i)5-s + (−0.580 − 0.814i)6-s + (0.654 + 0.755i)8-s + (0.235 + 0.971i)9-s + (0.786 − 0.618i)10-s + (−0.415 − 0.909i)11-s + (−0.327 − 0.945i)12-s + (−0.327 − 0.945i)13-s + (−0.959 + 0.281i)15-s + (0.415 + 0.909i)16-s + (0.888 + 0.458i)17-s + (−0.0475 + 0.998i)18-s + (0.959 − 0.281i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(469\)    =    \(7 \cdot 67\)
Sign: $0.663 - 0.747i$
Analytic conductor: \(2.17802\)
Root analytic conductor: \(2.17802\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{469} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 469,\ (0:\ ),\ 0.663 - 0.747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.915620737 - 0.8611057389i\)
\(L(\frac12)\) \(\approx\) \(1.915620737 - 0.8611057389i\)
\(L(1)\) \(\approx\) \(1.592541407 - 0.3169977745i\)
\(L(1)\) \(\approx\) \(1.592541407 - 0.3169977745i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.959 + 0.281i)T \)
3 \( 1 + (-0.786 - 0.618i)T \)
5 \( 1 + (0.580 - 0.814i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (-0.327 - 0.945i)T \)
17 \( 1 + (0.888 + 0.458i)T \)
19 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.888 - 0.458i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (0.786 + 0.618i)T \)
53 \( 1 + (0.888 - 0.458i)T \)
59 \( 1 + (-0.981 + 0.189i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.0475 - 0.998i)T \)
73 \( 1 + (0.995 - 0.0950i)T \)
79 \( 1 + (0.327 + 0.945i)T \)
83 \( 1 + (0.995 + 0.0950i)T \)
89 \( 1 + (0.786 - 0.618i)T \)
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

−23.46617451311684462604623955922, −23.12018980949688613844876230545, −22.06825090348758969483366313800, −21.80085995104025171079411018239, −20.78379099897927528256946362241, −20.15823586260621454428629738348, −18.62351559640745777179042962242, −18.18398004603202118279373388985, −16.80269560038731630959338396745, −16.23023579436200293725355423070, −15.043399907932146088627812967766, −14.565199984756869576467282429589, −13.63166625439719614851157231989, −12.40065333493436556216121711651, −11.81285906363890992263203530440, −10.80171837687790789231458512557, −10.10214408754787016083782332358, −9.44147167195996985852688067577, −7.28102978744581732076001843583, −6.71314683115284248218951437104, −5.58366512013766962293643926413, −4.95566621504508052217444318275, −3.83276143300959757992684509965, −2.797964978529918123166308616028, −1.57915397052914170117732975022, 1.01342349931866608289018663433, 2.268671234108154064147268948568, 3.5536329153075536605850550678, 5.03339880716994956980453933338, 5.563125210805746327276450505358, 6.16508963299504986190329828122, 7.599560227352659292576507007356, 8.08209815368719691788403626004, 9.72242944886494874778874297286, 10.84689084972190132173753730777, 11.75616294876994398558566365690, 12.53405566359283072242064843680, 13.3530537746346953081184249248, 13.74385217153453901525514729940, 15.122465023934013995627808202056, 16.130480640551139875231415897950, 16.79497708856347397756094798369, 17.459425908153550488902801742345, 18.46256838829610397711129192810, 19.66374529280163223545398991727, 20.5266396680705554279481374637, 21.5281197356495587280395884942, 22.03953644182130888955154210413, 23.01466846702284991184818752008, 23.8552792170641738466919456558

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