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Properties

Label 1-203-203.89-r0-0-0
Degree $1$
Conductor $203$
Sign $0.682 - 0.730i$
Analytic cond. $0.942728$
Root an. cond. $0.942728$
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  + (−0.294 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.682 - 0.730i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.682 - 0.730i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.294 - 0.955i)T \)
3 \( 1 + (0.997 + 0.0747i)T \)
5 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (0.149 + 0.988i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.997 + 0.0747i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
31 \( 1 + (0.680 - 0.733i)T \)
37 \( 1 + (0.149 - 0.988i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.974 + 0.222i)T \)
47 \( 1 + (-0.930 - 0.365i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.563 + 0.826i)T \)
67 \( 1 + (-0.365 - 0.930i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.294 - 0.955i)T \)
79 \( 1 + (-0.149 + 0.988i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.294 + 0.955i)T \)
97 \( 1 + (-0.433 + 0.900i)T \)
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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

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

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