Properties

Degree $1$
Conductor $203$
Sign $0.682 + 0.730i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.294 + 0.955i)2-s + (0.997 − 0.0747i)3-s + (−0.826 − 0.563i)4-s + (0.955 + 0.294i)5-s + (−0.222 + 0.974i)6-s + (0.781 − 0.623i)8-s + (0.988 − 0.149i)9-s + (−0.563 + 0.826i)10-s + (0.149 − 0.988i)11-s + (−0.866 − 0.5i)12-s + (0.623 − 0.781i)13-s + (0.974 + 0.222i)15-s + (0.365 + 0.930i)16-s + (−0.866 + 0.5i)17-s + (−0.149 + 0.988i)18-s + (−0.997 − 0.0747i)19-s + ⋯
L(s,χ)  = 1  + (−0.294 + 0.955i)2-s + (0.997 − 0.0747i)3-s + (−0.826 − 0.563i)4-s + (0.955 + 0.294i)5-s + (−0.222 + 0.974i)6-s + (0.781 − 0.623i)8-s + (0.988 − 0.149i)9-s + (−0.563 + 0.826i)10-s + (0.149 − 0.988i)11-s + (−0.866 − 0.5i)12-s + (0.623 − 0.781i)13-s + (0.974 + 0.222i)15-s + (0.365 + 0.930i)16-s + (−0.866 + 0.5i)17-s + (−0.149 + 0.988i)18-s + (−0.997 − 0.0747i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.682 + 0.730i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.682 + 0.730i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.682 + 0.730i$
Motivic weight: \(0\)
Character: $\chi_{203} (73, \cdot )$
Sato-Tate group: $\mu(84)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.682 + 0.730i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.394701203 + 0.6057302887i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.394701203 + 0.6057302887i\)
\(L(\chi,1)\) \(\approx\) \(1.252218555 + 0.4509994891i\)
\(L(1,\chi)\) \(\approx\) \(1.252218555 + 0.4509994891i\)

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

−26.62975561036306291459010652850, −25.84038836435345750577246817651, −25.2565739053028475159416132247, −23.98078612061292878500263703173, −22.6096325303922820120306858223, −21.550622125984969837140109094854, −20.94678865949383582655136306022, −20.17554238321391935004036097612, −19.29720988972702344238331255277, −18.22752902406830174212446013882, −17.49091827035524840110431902041, −16.24107272669523825402268309404, −14.821458202170119224064041523312, −13.7402723502191503254807300297, −13.20924900788527347229301314502, −12.1200993221508203350991492312, −10.71581825531257649081894956995, −9.64747249009366702390750599322, −9.14356183986715658958847344643, −8.09248078298242769549527191919, −6.660507433292451268849081648663, −4.78616284132112497282657226288, −3.86101738252133583142822063695, −2.289679205632613569006920533944, −1.73457927752881451779320409994, 1.50171297102189393662942612817, 3.04495256979581971455113379268, 4.4837199701375468964452535413, 6.03746022671520686744136046335, 6.67541463206020673277411213985, 8.26020384532673085080896913592, 8.66864149893119779532069291039, 9.915981798406960362182238809147, 10.71858956005995423827455203326, 12.98686871668949016298980891824, 13.54211193101045139389989634395, 14.44468230177704393409106126683, 15.25810844203269919731586597034, 16.29337743197115313215961461480, 17.45599049314526390288657460076, 18.3104568832254614254728695818, 19.10961711327543567345714719482, 20.17662864600031912519730049702, 21.44665367987037434359984721104, 22.15918508749746268504938215196, 23.512957852048850343487421103308, 24.552289208601426299103574675252, 25.103608549618467581866280660778, 26.00885207504992093055026981831, 26.55145509937661542608872363303

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