Properties

Degree $1$
Conductor $283$
Sign $0.999 - 0.0320i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.999 - 0.0320i$
Motivic weight: \(0\)
Character: $\chi_{283} (149, \cdot )$
Sato-Tate group: $\mu(94)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.999 - 0.0320i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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