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Properties

Label 1-427-427.213-r0-0-0
Degree $1$
Conductor $427$
Sign $0.182 - 0.983i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.406 − 0.913i)2-s + (0.913 + 0.406i)3-s + (−0.669 − 0.743i)4-s + (−0.978 + 0.207i)5-s + (0.743 − 0.669i)6-s + (−0.951 + 0.309i)8-s + (0.669 + 0.743i)9-s + (−0.207 + 0.978i)10-s + (0.866 − 0.5i)11-s + (−0.309 − 0.951i)12-s + (0.5 − 0.866i)13-s + (−0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (0.743 − 0.669i)17-s + (0.951 − 0.309i)18-s + (−0.809 + 0.587i)19-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (0.913 + 0.406i)3-s + (−0.669 − 0.743i)4-s + (−0.978 + 0.207i)5-s + (0.743 − 0.669i)6-s + (−0.951 + 0.309i)8-s + (0.669 + 0.743i)9-s + (−0.207 + 0.978i)10-s + (0.866 − 0.5i)11-s + (−0.309 − 0.951i)12-s + (0.5 − 0.866i)13-s + (−0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (0.743 − 0.669i)17-s + (0.951 − 0.309i)18-s + (−0.809 + 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 427 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(427\)    =    \(7 \cdot 61\)
Sign: $0.182 - 0.983i$
Analytic conductor: \(1.98297\)
Root analytic conductor: \(1.98297\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{427} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 427,\ (0:\ ),\ 0.182 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.412248003 - 1.174069898i\)
\(L(\frac12)\) \(\approx\) \(1.412248003 - 1.174069898i\)
\(L(1)\) \(\approx\) \(1.317636305 - 0.6364519631i\)
\(L(1)\) \(\approx\) \(1.317636305 - 0.6364519631i\)

Euler product

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

−24.334723084298197671992239843140, −23.4584199895489110597413135067, −23.217436461433145835184981704269, −21.70112315805614217667620046390, −21.09568040098803815708983909436, −19.82364669912833977543658369640, −19.2867913642810164701373457517, −18.317755714639278436509872175503, −17.25332095230076765784447459326, −16.32299181032636481142167284370, −15.44293037565318562267282827026, −14.69507050623334606348429501870, −14.1065969778581087412144101448, −12.89442026741181962081481152554, −12.38539507857723441015292042493, −11.301904840079340914607136899, −9.515307890146937684150124459115, −8.74451041645294987862129886721, −8.04935275982268152484027335208, −7.03755835575865291262811414973, −6.47628206636545477649592906717, −4.78575855541591791682704561400, −3.943215398652029426928692208987, −3.17954009644157047436984513640, −1.40896995245953794205264824361, 0.998972389321233089548555362636, 2.55944300971777899039627769141, 3.46231466487181776186279080272, 4.045381275451834844595723473719, 5.16163671744112330457943043941, 6.60211431909825865165578819641, 8.079379648459634494928630778496, 8.645989747795120696904268170392, 9.791244461859618660897203996553, 10.6422865264963123517987646119, 11.49898070861100438890112766419, 12.45556274583624660579168073919, 13.39313123338304343060482778028, 14.37337176287226933338877339355, 14.93982928949356426278964382461, 15.80311285495761263445108584756, 16.92722727871159008877441078749, 18.571253433041018147345602361656, 18.95447907800262951132562152506, 19.810312194860731669916658169761, 20.51478489025562707596732688917, 21.16009527254431258294713442293, 22.29138405010201305927932115080, 22.82342767297944353770908626154, 23.815430996061961399115825326605

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