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Properties

Label 1-283-283.19-r1-0-0
Degree $1$
Conductor $283$
Sign $0.999 + 0.0320i$
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  + (−0.231 − 0.972i)2-s + (−0.100 − 0.994i)3-s + (−0.892 + 0.451i)4-s + (0.979 + 0.199i)5-s + (−0.944 + 0.328i)6-s + (0.920 + 0.390i)7-s + (0.645 + 0.763i)8-s + (−0.979 + 0.199i)9-s + (−0.0334 − 0.999i)10-s + (−0.892 − 0.451i)11-s + (0.538 + 0.842i)12-s + (−0.944 + 0.328i)13-s + (0.166 − 0.986i)14-s + (0.100 − 0.994i)15-s + (0.593 − 0.805i)16-s + (0.166 + 0.986i)17-s + ⋯
L(s)  = 1  + (−0.231 − 0.972i)2-s + (−0.100 − 0.994i)3-s + (−0.892 + 0.451i)4-s + (0.979 + 0.199i)5-s + (−0.944 + 0.328i)6-s + (0.920 + 0.390i)7-s + (0.645 + 0.763i)8-s + (−0.979 + 0.199i)9-s + (−0.0334 − 0.999i)10-s + (−0.892 − 0.451i)11-s + (0.538 + 0.842i)12-s + (−0.944 + 0.328i)13-s + (0.166 − 0.986i)14-s + (0.100 − 0.994i)15-s + (0.593 − 0.805i)16-s + (0.166 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.999 + 0.0320i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.999 + 0.0320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210580815 + 0.01943086371i\)
\(L(\frac12)\) \(\approx\) \(1.210580815 + 0.01943086371i\)
\(L(1)\) \(\approx\) \(0.8289800500 - 0.4087851624i\)
\(L(1)\) \(\approx\) \(0.8289800500 - 0.4087851624i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.231 - 0.972i)T \)
3 \( 1 + (-0.100 - 0.994i)T \)
5 \( 1 + (0.979 + 0.199i)T \)
7 \( 1 + (0.920 + 0.390i)T \)
11 \( 1 + (-0.892 - 0.451i)T \)
13 \( 1 + (-0.944 + 0.328i)T \)
17 \( 1 + (0.166 + 0.986i)T \)
19 \( 1 + (0.944 - 0.328i)T \)
23 \( 1 + (-0.824 + 0.565i)T \)
29 \( 1 + (-0.420 + 0.907i)T \)
31 \( 1 + (-0.964 + 0.264i)T \)
37 \( 1 + (-0.359 + 0.933i)T \)
41 \( 1 + (-0.645 + 0.763i)T \)
43 \( 1 + (0.741 + 0.670i)T \)
47 \( 1 + (0.944 + 0.328i)T \)
53 \( 1 + (-0.593 - 0.805i)T \)
59 \( 1 + (0.695 + 0.718i)T \)
61 \( 1 + (-0.0334 + 0.999i)T \)
67 \( 1 + (0.538 - 0.842i)T \)
71 \( 1 + (-0.892 - 0.451i)T \)
73 \( 1 + (0.964 + 0.264i)T \)
79 \( 1 + (0.741 - 0.670i)T \)
83 \( 1 + (-0.296 - 0.955i)T \)
89 \( 1 + (-0.741 - 0.670i)T \)
97 \( 1 + (-0.824 + 0.565i)T \)
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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

−25.359387088586562430620500766288, −24.609459626396809108181082890863, −23.67501932322820723975737383759, −22.558938285532170039322880420346, −21.96004818905697078940034792012, −20.74350498927823099638140242587, −20.28985205626459403303502032054, −18.492158634589927324510080778024, −17.70331630257477172503132778088, −17.09020399491837325330444327876, −16.191238996766351049442939509, −15.28006338841643322700956501352, −14.2801261919515813413357736013, −13.8266173235204200186740585806, −12.36048948106049901003600000794, −10.78816988761863246310889303381, −9.96169069123391935704952471360, −9.35168287320338487184733008560, −8.10734512772702894719493770013, −7.193854123818361258622259132813, −5.447549521620680652724832160386, −5.2982348498155609392181668754, −4.134394466524772878975086378264, −2.284201479105081561270718963747, −0.40825076334339522936258164689, 1.34492678848282397921382705113, 2.09667461690596063399340046428, 3.07171786299967818987519317267, 5.02685326174715679501988215776, 5.73023250363654844627672287237, 7.35357906832038395117785847912, 8.239523715732708945262208184, 9.26457722440058899847360694647, 10.43229166171495658224289312376, 11.3178349958391025842569319467, 12.2433335748817359980290026290, 13.123568686687359613106508393201, 13.95105801175561572688822661918, 14.69128849227353129091781810032, 16.66616648535386701943151020169, 17.61811700533637384628186423812, 18.1056168815149331456030660616, 18.825577515288326409778949995574, 19.84938876583624713341818942981, 20.81704496078246798581108647651, 21.75950032464584461309529191936, 22.25023828932045590504064284105, 23.79693848899136160801190372163, 24.20703154134795210523165814839, 25.5231096622062397649636293313

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