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Properties

Label 1-283-283.27-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.545 - 0.837i$
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  + (−0.359 − 0.933i)2-s + (0.997 + 0.0667i)3-s + (−0.741 + 0.670i)4-s + (−0.991 + 0.133i)5-s + (−0.296 − 0.955i)6-s + (0.964 − 0.264i)7-s + (0.892 + 0.451i)8-s + (0.991 + 0.133i)9-s + (0.480 + 0.876i)10-s + (−0.741 − 0.670i)11-s + (−0.784 + 0.619i)12-s + (−0.296 − 0.955i)13-s + (−0.593 − 0.805i)14-s + (−0.997 + 0.0667i)15-s + (0.100 − 0.994i)16-s + (−0.593 + 0.805i)17-s + ⋯
L(s)  = 1  + (−0.359 − 0.933i)2-s + (0.997 + 0.0667i)3-s + (−0.741 + 0.670i)4-s + (−0.991 + 0.133i)5-s + (−0.296 − 0.955i)6-s + (0.964 − 0.264i)7-s + (0.892 + 0.451i)8-s + (0.991 + 0.133i)9-s + (0.480 + 0.876i)10-s + (−0.741 − 0.670i)11-s + (−0.784 + 0.619i)12-s + (−0.296 − 0.955i)13-s + (−0.593 − 0.805i)14-s + (−0.997 + 0.0667i)15-s + (0.100 − 0.994i)16-s + (−0.593 + 0.805i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.545 - 0.837i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.545 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7915870456 - 1.460186661i\)
\(L(\frac12)\) \(\approx\) \(0.7915870456 - 1.460186661i\)
\(L(1)\) \(\approx\) \(0.9295292051 - 0.5416435825i\)
\(L(1)\) \(\approx\) \(0.9295292051 - 0.5416435825i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.359 - 0.933i)T \)
3 \( 1 + (0.997 + 0.0667i)T \)
5 \( 1 + (-0.991 + 0.133i)T \)
7 \( 1 + (0.964 - 0.264i)T \)
11 \( 1 + (-0.741 - 0.670i)T \)
13 \( 1 + (-0.296 - 0.955i)T \)
17 \( 1 + (-0.593 + 0.805i)T \)
19 \( 1 + (0.296 + 0.955i)T \)
23 \( 1 + (0.920 + 0.390i)T \)
29 \( 1 + (0.231 - 0.972i)T \)
31 \( 1 + (0.645 - 0.763i)T \)
37 \( 1 + (-0.695 - 0.718i)T \)
41 \( 1 + (-0.892 + 0.451i)T \)
43 \( 1 + (0.0334 - 0.999i)T \)
47 \( 1 + (0.296 - 0.955i)T \)
53 \( 1 + (-0.100 - 0.994i)T \)
59 \( 1 + (0.860 - 0.509i)T \)
61 \( 1 + (0.480 - 0.876i)T \)
67 \( 1 + (-0.784 - 0.619i)T \)
71 \( 1 + (-0.741 - 0.670i)T \)
73 \( 1 + (-0.645 - 0.763i)T \)
79 \( 1 + (0.0334 + 0.999i)T \)
83 \( 1 + (-0.979 - 0.199i)T \)
89 \( 1 + (-0.0334 + 0.999i)T \)
97 \( 1 + (0.920 + 0.390i)T \)
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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

−25.68138197123436035287920058433, −24.69299298289119230834502576158, −24.06557295693370123140433998917, −23.44023206980513081824455835636, −22.165407122084805006367913257533, −20.918032216586641405062200342057, −20.07380981530272256438159458253, −19.14233310914949909157306647032, −18.39897455323600210182273275606, −17.523810938565926115116937397446, −16.122338218659781658383433100521, −15.48509118180079700637813390712, −14.7655554020141817683641580299, −13.94701832029625280738567481517, −12.86413117416544805650747309881, −11.588577142939136692940317899538, −10.33245781152630836976533263441, −8.95503052277078472708495034547, −8.59196255709400112944620672374, −7.33433098714339706898119353585, −7.055729652430038531851973173663, −4.80696571443304222125189454530, −4.57826981163589280704157075632, −2.76564397364609386727232454278, −1.253309451979001126177751152620, 0.56618716050078562431044252552, 1.997974008842675997634833907598, 3.1867597274346571054370184537, 3.97154141650764342205448400929, 5.08526625001946901664232064997, 7.40351966063142921490864831084, 8.135772952908920466417585529855, 8.5690285991069168252819871215, 10.109451630510135638367046656541, 10.79393256750264270192997624589, 11.795298040952139231410909803550, 12.902470934805438420347187356459, 13.72261485633360151443696126720, 14.84047708283515842398482870362, 15.6132325838985486977071145988, 16.96127862772445584231066650548, 18.076421424704415658034375627460, 18.9620005386768905077208026344, 19.56740510048964086258585185394, 20.57091702631199451405585030240, 20.96780824795200443043171178539, 22.07644796816891835875887021672, 23.19915546871484164520389114112, 24.14908212958611696122297823729, 25.14260899572385051316010736478

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