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Properties

Label 1-283-283.69-r1-0-0
Degree $1$
Conductor $283$
Sign $0.981 + 0.192i$
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  + (0.944 − 0.328i)2-s + (−0.628 + 0.777i)3-s + (0.784 − 0.619i)4-s + (−0.951 + 0.306i)5-s + (−0.338 + 0.940i)6-s + (−0.911 − 0.410i)7-s + (0.538 − 0.842i)8-s + (−0.210 − 0.977i)9-s + (−0.798 + 0.602i)10-s + (−0.929 + 0.369i)11-s + (−0.0111 + 0.999i)12-s + (0.984 − 0.177i)13-s + (−0.996 − 0.0890i)14-s + (0.359 − 0.933i)15-s + (0.231 − 0.972i)16-s + (0.575 + 0.818i)17-s + ⋯
L(s)  = 1  + (0.944 − 0.328i)2-s + (−0.628 + 0.777i)3-s + (0.784 − 0.619i)4-s + (−0.951 + 0.306i)5-s + (−0.338 + 0.940i)6-s + (−0.911 − 0.410i)7-s + (0.538 − 0.842i)8-s + (−0.210 − 0.977i)9-s + (−0.798 + 0.602i)10-s + (−0.929 + 0.369i)11-s + (−0.0111 + 0.999i)12-s + (0.984 − 0.177i)13-s + (−0.996 − 0.0890i)14-s + (0.359 − 0.933i)15-s + (0.231 − 0.972i)16-s + (0.575 + 0.818i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.981 + 0.192i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.981 + 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.013215744 + 0.1952166751i\)
\(L(\frac12)\) \(\approx\) \(2.013215744 + 0.1952166751i\)
\(L(1)\) \(\approx\) \(1.273161893 + 0.01994388714i\)
\(L(1)\) \(\approx\) \(1.273161893 + 0.01994388714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.944 - 0.328i)T \)
3 \( 1 + (-0.628 + 0.777i)T \)
5 \( 1 + (-0.951 + 0.306i)T \)
7 \( 1 + (-0.911 - 0.410i)T \)
11 \( 1 + (-0.929 + 0.369i)T \)
13 \( 1 + (0.984 - 0.177i)T \)
17 \( 1 + (0.575 + 0.818i)T \)
19 \( 1 + (0.645 + 0.763i)T \)
23 \( 1 + (0.400 - 0.916i)T \)
29 \( 1 + (0.480 + 0.876i)T \)
31 \( 1 + (-0.441 - 0.897i)T \)
37 \( 1 + (0.679 + 0.734i)T \)
41 \( 1 + (-0.460 + 0.887i)T \)
43 \( 1 + (0.824 + 0.565i)T \)
47 \( 1 + (0.338 + 0.940i)T \)
53 \( 1 + (-0.231 - 0.972i)T \)
59 \( 1 + (0.317 - 0.948i)T \)
61 \( 1 + (0.920 - 0.390i)T \)
67 \( 1 + (-0.860 + 0.509i)T \)
71 \( 1 + (0.784 + 0.619i)T \)
73 \( 1 + (0.441 - 0.897i)T \)
79 \( 1 + (0.824 - 0.565i)T \)
83 \( 1 + (0.0556 + 0.998i)T \)
89 \( 1 + (0.902 - 0.431i)T \)
97 \( 1 + (-0.993 + 0.111i)T \)
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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

−25.11179995550185478555837471713, −24.15919883595539253332661489233, −23.33416819744984383664373490826, −23.041552431119405150745306154041, −22.004367038633512997592572959, −20.961361986140446695264715446810, −19.83607196627947706237398578183, −19.00223727217792765872281033257, −18.053997282916595838808566797946, −16.68867712699683976331841807885, −15.93205936302133197243149313946, −15.54043024479940093059800203518, −13.801415267545355427767387621313, −13.22118788225853379737535856092, −12.296509472668386828655023997818, −11.621655473041561840778840147828, −10.73719301149121624223780222496, −8.82979534383737796055638013256, −7.66685990915616356475645986352, −6.99625202212918392071439167846, −5.79265878888762305227875776918, −5.10401360765662120990289730137, −3.62093306162121035289155924293, −2.620601788011102216211966457908, −0.72842739058681097802636939680, 0.821461392869915017000319406902, 3.05132105094402338499196899240, 3.69736478809928668009301330891, 4.65798973401839146417122070881, 5.86103103259672063455243628879, 6.710132129312491806463161938887, 7.999383321336329200402608266942, 9.78439779539344456574820693143, 10.56319475270863895322429838170, 11.21116745173001148302162842849, 12.37724535929570382555867045260, 12.98006698950522170275157437416, 14.43946082255380986442968286059, 15.26234778304265626523232104315, 16.10061053225985084341390023535, 16.52418343086353322396198891355, 18.28336126320307090402650761924, 19.1561780920468750157545772799, 20.37798087664007082578666524285, 20.7552655497934936134704866048, 22.050550892163031241006730614475, 22.70668285032886803927247753988, 23.38435864331965998689602929681, 23.81444497729961631546722030527, 25.506733893584120738402208873962

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