↑

Properties

Label 1-427-427.317-r0-0-0
Degree $1$
Conductor $427$
Sign $0.479 + 0.877i$
Analytic cond. $1.98297$
Root an. cond. $1.98297$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)2-s + (0.669 + 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.913 − 0.406i)6-s + (−0.809 + 0.587i)8-s + (−0.104 + 0.994i)9-s + (−0.809 + 0.587i)10-s + (−0.5 + 0.866i)11-s + (−0.104 − 0.994i)12-s + (−0.5 + 0.866i)13-s + (−0.104 − 0.994i)15-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)17-s + (0.913 + 0.406i)18-s + (0.669 + 0.743i)19-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (0.669 + 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.913 − 0.406i)6-s + (−0.809 + 0.587i)8-s + (−0.104 + 0.994i)9-s + (−0.809 + 0.587i)10-s + (−0.5 + 0.866i)11-s + (−0.104 − 0.994i)12-s + (−0.5 + 0.866i)13-s + (−0.104 − 0.994i)15-s + (0.309 + 0.951i)16-s + (−0.809 − 0.587i)17-s + (0.913 + 0.406i)18-s + (0.669 + 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(427\)    =    \(7 \cdot 61\)
Sign: $0.479 + 0.877i$
Analytic conductor: \(1.98297\)
Root analytic conductor: \(1.98297\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{427} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 427,\ (0:\ ),\ 0.479 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7891738031 + 0.4680315668i\)
\(L(\frac12)\) \(\approx\) \(0.7891738031 + 0.4680315668i\)
\(L(1)\) \(\approx\) \(0.9823375743 - 0.08689496994i\)
\(L(1)\) \(\approx\) \(0.9823375743 - 0.08689496994i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.104 - 0.994i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.33414181596402463557469345423, −23.39042808646205598368690918521, −22.49143920774861240097939227470, −21.76436981847189193736362746105, −20.40568754550245224517154742001, −19.60640619060081744894069015274, −18.58299106669751982131209244413, −18.13992547643794407101501974126, −17.043886824340027856281325927954, −15.86331473452705695127173546000, −15.16684836841808122840861272305, −14.52247425469868133027054342232, −13.51169693458618732029144646940, −12.85655711951540594387734423625, −11.87099032282561240338835338752, −10.68560507240130631242901523880, −9.22445073182393453509638987731, −8.23887674740002351775221713600, −7.72962381738884034710854015574, −6.81676231817105843287829468805, −5.95927137039984180861075273515, −4.58442375765771067206419585639, −3.37458600933825123069762068386, −2.68891526057081563352143745981, −0.43041358263724150991208063333, 1.69001218409829025702178331796, 2.77852447214831767247391022241, 3.92669951400213640228374304000, 4.56249900493759562000903517048, 5.37453114412634955713510613347, 7.33448912584407830141934971384, 8.30970575146724343038947300700, 9.38950225718730422444250108069, 9.808765659022072790804316032983, 11.097650520758118149945384222183, 11.804022474344344663633824131250, 12.78888799222974481316127126648, 13.701525460601740869223330485, 14.60094095553704915134571398305, 15.46882482916676172375496682607, 16.20615441959930724318243097310, 17.45094228510452531986126649859, 18.61942476321945473001784429005, 19.55150507760965253748976796606, 20.05732413923627605933604441421, 20.80969444692097885419368860917, 21.46999807366004552435011733179, 22.51136214154342631505554018105, 23.19417911724730166062233354345, 24.18921249658117360704360370611

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