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  :  $\chi_{53} (20, \cdot )$
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

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

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