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Properties

Label 1-283-283.223-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.930 + 0.365i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (−0.100 − 0.994i)2-s + (−0.964 + 0.264i)3-s + (−0.979 + 0.199i)4-s + (−0.860 − 0.509i)5-s + (0.359 + 0.933i)6-s + (0.480 + 0.876i)7-s + (0.296 + 0.955i)8-s + (0.860 − 0.509i)9-s + (−0.420 + 0.907i)10-s + (−0.979 − 0.199i)11-s + (0.892 − 0.451i)12-s + (0.359 + 0.933i)13-s + (0.824 − 0.565i)14-s + (0.964 + 0.264i)15-s + (0.920 − 0.390i)16-s + (0.824 + 0.565i)17-s + ⋯
L(s)  = 1  + (−0.100 − 0.994i)2-s + (−0.964 + 0.264i)3-s + (−0.979 + 0.199i)4-s + (−0.860 − 0.509i)5-s + (0.359 + 0.933i)6-s + (0.480 + 0.876i)7-s + (0.296 + 0.955i)8-s + (0.860 − 0.509i)9-s + (−0.420 + 0.907i)10-s + (−0.979 − 0.199i)11-s + (0.892 − 0.451i)12-s + (0.359 + 0.933i)13-s + (0.824 − 0.565i)14-s + (0.964 + 0.264i)15-s + (0.920 − 0.390i)16-s + (0.824 + 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.930 + 0.365i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.930 + 0.365i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02802979097 - 0.1478637868i\)
\(L(\frac12)\) \(\approx\) \(0.02802979097 - 0.1478637868i\)
\(L(1)\) \(\approx\) \(0.5029483587 - 0.1923089973i\)
\(L(1)\) \(\approx\) \(0.5029483587 - 0.1923089973i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.100 - 0.994i)T \)
3 \( 1 + (-0.964 + 0.264i)T \)
5 \( 1 + (-0.860 - 0.509i)T \)
7 \( 1 + (0.480 + 0.876i)T \)
11 \( 1 + (-0.979 - 0.199i)T \)
13 \( 1 + (0.359 + 0.933i)T \)
17 \( 1 + (0.824 + 0.565i)T \)
19 \( 1 + (-0.359 - 0.933i)T \)
23 \( 1 + (-0.0334 - 0.999i)T \)
29 \( 1 + (0.593 - 0.805i)T \)
31 \( 1 + (0.944 + 0.328i)T \)
37 \( 1 + (0.997 - 0.0667i)T \)
41 \( 1 + (-0.296 + 0.955i)T \)
43 \( 1 + (-0.991 + 0.133i)T \)
47 \( 1 + (-0.359 + 0.933i)T \)
53 \( 1 + (-0.920 - 0.390i)T \)
59 \( 1 + (-0.538 + 0.842i)T \)
61 \( 1 + (-0.420 - 0.907i)T \)
67 \( 1 + (0.892 + 0.451i)T \)
71 \( 1 + (-0.979 - 0.199i)T \)
73 \( 1 + (-0.944 + 0.328i)T \)
79 \( 1 + (-0.991 - 0.133i)T \)
83 \( 1 + (0.695 - 0.718i)T \)
89 \( 1 + (0.991 - 0.133i)T \)
97 \( 1 + (-0.0334 - 0.999i)T \)
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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

−25.82187528819827051076704877529, −24.863502542040656538488593589407, −23.67514375350114723789435222525, −23.305890084697981988105347301146, −22.870973771431543178635860117632, −21.65532254585787781995329585587, −20.383860641702491693447332767297, −19.04294484653529225587253635863, −18.30483576231906707656445743616, −17.61493030452415432579200341523, −16.613337535331275155288164829, −15.85463299634480873260939745725, −15.03178452884481984308193270296, −13.87775141169965583142290633501, −12.92685094615872758446089818618, −11.8369792018236855715414078386, −10.59627309090948676314529803099, −10.114796268462754064847705542682, −8.00830651086364336274738485179, −7.71683253414211011853526155313, −6.73178902543231413070505435365, −5.54931544005280686606413277627, −4.67760108451826060215462974235, −3.492115549600735751633123924513, −1.05713206816733147714053789021, 0.0724665076941418496812161912, 1.35481561468474297263564914242, 2.901589350041646856012485279701, 4.41492967900447001690712501362, 4.8745322771825141510544125364, 6.16169434627334415472667761927, 7.93764927076727515371472299663, 8.71787580769284960172855053875, 9.89798380141013756498111315850, 11.01300333039470066693502705152, 11.61814721666819424644316458448, 12.36932393407207519025808312264, 13.1794034367325428041008916902, 14.73744375654267298472256916085, 15.77181272580969392863786941279, 16.656044737583554464110232429078, 17.68677533754114344556477486024, 18.64522826800908340219615898285, 19.16306144584097655875102632159, 20.560448990824241931138225989273, 21.31983138807732045599114413509, 21.810018019023610392365331335372, 23.179211894626422987219610555170, 23.53499439806035698041878840411, 24.502598813540191974667935072

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