Properties

Label 1-407-407.114-r0-0-0
Degree $1$
Conductor $407$
Sign $0.335 - 0.941i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (0.882 − 0.469i)2-s + (−0.719 + 0.694i)3-s + (0.559 − 0.829i)4-s + (−0.848 + 0.529i)5-s + (−0.309 + 0.951i)6-s + (−0.997 − 0.0697i)7-s + (0.104 − 0.994i)8-s + (0.0348 − 0.999i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (0.615 + 0.788i)13-s + (−0.913 + 0.406i)14-s + (0.241 − 0.970i)15-s + (−0.374 − 0.927i)16-s + (0.615 − 0.788i)17-s + (−0.438 − 0.898i)18-s + ⋯
L(s)  = 1  + (0.882 − 0.469i)2-s + (−0.719 + 0.694i)3-s + (0.559 − 0.829i)4-s + (−0.848 + 0.529i)5-s + (−0.309 + 0.951i)6-s + (−0.997 − 0.0697i)7-s + (0.104 − 0.994i)8-s + (0.0348 − 0.999i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (0.615 + 0.788i)13-s + (−0.913 + 0.406i)14-s + (0.241 − 0.970i)15-s + (−0.374 − 0.927i)16-s + (0.615 − 0.788i)17-s + (−0.438 − 0.898i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.335 - 0.941i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.335 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9947271713 - 0.7013754502i\)
\(L(\frac12)\) \(\approx\) \(0.9947271713 - 0.7013754502i\)
\(L(1)\) \(\approx\) \(1.082272382 - 0.2558865924i\)
\(L(1)\) \(\approx\) \(1.082272382 - 0.2558865924i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.882 - 0.469i)T \)
3 \( 1 + (-0.719 + 0.694i)T \)
5 \( 1 + (-0.848 + 0.529i)T \)
7 \( 1 + (-0.997 - 0.0697i)T \)
13 \( 1 + (0.615 + 0.788i)T \)
17 \( 1 + (0.615 - 0.788i)T \)
19 \( 1 + (0.719 - 0.694i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.913 - 0.406i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.961 + 0.275i)T \)
43 \( 1 - T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (0.848 + 0.529i)T \)
59 \( 1 + (0.241 - 0.970i)T \)
61 \( 1 + (-0.0348 - 0.999i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.882 - 0.469i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.374 - 0.927i)T \)
83 \( 1 + (-0.615 + 0.788i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−24.28279240266350847394920983503, −23.461718169792623878763856954858, −22.974179230814900697421630538730, −22.33417735753143670120488336182, −21.22877840339293405281204157975, −20.11642762355081103788205085254, −19.39200164138312047523957788676, −18.35196522707835918414439178662, −17.200706429564897851911652348356, −16.39644618043032395327893953531, −15.89562931596375097289765953968, −14.90270985995101940504744157026, −13.56090788381962086701227967039, −12.84220020970806822095470022371, −12.32629371761004157041226120355, −11.46540540749037935191268493525, −10.4364275972445066904729492444, −8.74358940212983335252989762489, −7.713969390654826392623408551043, −7.080795304774737400439355074408, −5.83402725009906885766020422429, −5.38428926418000538741673157273, −3.92006016117211473965322875039, −3.12431609920602649025819956062, −1.317324042869435460010002201284, 0.64741098290959353711236596010, 2.75210558533218643289875096645, 3.63719644793176247248323877115, 4.35545009118027432596190719154, 5.52965733707616777811869158246, 6.52133318954867267122910388986, 7.2282590568829077420135578886, 9.186564903357553849338200860250, 9.97781323003613194214016641282, 11.021482898686655832911645293391, 11.55058076374919021018329291121, 12.38010539432542404625454137573, 13.43555934176330860521794575408, 14.52718346807994987322492649949, 15.39938441024519212884699833114, 16.08914671931157861545404329702, 16.68849696220804357742131989789, 18.45336397650423020038646275048, 18.92241363493285995029560626352, 20.1034240129432852514453395571, 20.73727766405011006523309063257, 21.88698081887932219542016694297, 22.4240739870005749130811487950, 23.13656475246857990877282720868, 23.61038354588539539113576511967

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