Properties

Degree 1
Conductor 53
Sign $0.997 + 0.0685i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.935 + 0.354i)2-s + (0.992 − 0.120i)3-s + (0.748 + 0.663i)4-s + (−0.239 − 0.970i)5-s + (0.970 + 0.239i)6-s + (0.354 − 0.935i)7-s + (0.464 + 0.885i)8-s + (0.970 − 0.239i)9-s + (0.120 − 0.992i)10-s + (−0.568 + 0.822i)11-s + (0.822 + 0.568i)12-s + (−0.748 + 0.663i)13-s + (0.663 − 0.748i)14-s + (−0.354 − 0.935i)15-s + (0.120 + 0.992i)16-s + (−0.885 − 0.464i)17-s + ⋯
L(s,χ)  = 1  + (0.935 + 0.354i)2-s + (0.992 − 0.120i)3-s + (0.748 + 0.663i)4-s + (−0.239 − 0.970i)5-s + (0.970 + 0.239i)6-s + (0.354 − 0.935i)7-s + (0.464 + 0.885i)8-s + (0.970 − 0.239i)9-s + (0.120 − 0.992i)10-s + (−0.568 + 0.822i)11-s + (0.822 + 0.568i)12-s + (−0.748 + 0.663i)13-s + (0.663 − 0.748i)14-s + (−0.354 − 0.935i)15-s + (0.120 + 0.992i)16-s + (−0.885 − 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.997 + 0.0685i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.997 + 0.0685i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(53\)
\( \varepsilon \)  =  $0.997 + 0.0685i$
motivic weight  =  \(0\)
character  :  $\chi_{53} (8, \cdot )$
Sato-Tate  :  $\mu(52)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 53,\ (1:\ ),\ 0.997 + 0.0685i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.398935701 + 0.1166999320i$
$L(\frac12,\chi)$  $\approx$  $3.398935701 + 0.1166999320i$
$L(\chi,1)$  $\approx$  2.303288861 + 0.1132617516i
$L(1,\chi)$  $\approx$  2.303288861 + 0.1132617516i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−32.705969443808085090768835622951, −31.7103069658313842453840313780, −30.896810297462069029238302900769, −30.1435865522640938478737143398, −28.821806367141590312092691334043, −27.26176788893948490577033530456, −26.13590490445421324986311315222, −24.821100177624043304785062436632, −24.01062777541410644414117743570, −22.218529184770917826287635666520, −21.728302751881387527867399340512, −20.36334450761878011958893044602, −19.26191358864989120639541270766, −18.323948602591788745284863762049, −15.71209764123208505486206494514, −15.058349595473160509330468354842, −14.06445534109558183265536982177, −12.82374604979849407707636419045, −11.31774483967039799784121607668, −10.128178026851173458497170809035, −8.31733182302766345410589325731, −6.77266646310682236437119662921, −5.020940802055195509470665923191, −3.22452092238083069635378367436, −2.38715394433710642836174020563, 1.97289795196752793674060497126, 3.92877855102882561553832800766, 4.89378173269170792366775769296, 7.18586491694581651241347212714, 7.988752158853460092607275414214, 9.689559067388522909880805716, 11.75209816871288466438297988511, 13.04422539724473121877352767845, 13.85201626577344961497781314128, 15.108764253189463930706090014507, 16.21459360837240778388280118499, 17.52206970661628208552305545348, 19.65346258766410801419687670046, 20.48082569292972535585649480173, 21.20344001358278173366365335494, 22.9686624166951325958625250889, 24.12377131480185969848164919269, 24.72224320533929349332715637774, 26.063509021722378088059591464120, 27.02299717584110313749938145761, 28.884966796669080444170240869618, 30.03960144342662431570410604027, 31.30005209483440627706538510303, 31.65496936143035485907024556326, 33.01161232914738152718153371571

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