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Properties

Label 1-53-53.20-r1-0-0
Degree $1$
Conductor $53$
Sign $0.997 - 0.0685i$
Analytic cond. $5.69564$
Root an. cond. $5.69564$
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  + (0.935 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.997 - 0.0685i$
Analytic conductor: \(5.69564\)
Root analytic conductor: \(5.69564\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (1:\ ),\ 0.997 - 0.0685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.398935701 - 0.1166999320i\)
\(L(\frac12)\) \(\approx\) \(3.398935701 - 0.1166999320i\)
\(L(1)\) \(\approx\) \(2.303288861 - 0.1132617516i\)
\(L(1)\) \(\approx\) \(2.303288861 - 0.1132617516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
good2 \( 1 + (0.935 - 0.354i)T \)
3 \( 1 + (0.992 + 0.120i)T \)
5 \( 1 + (-0.239 + 0.970i)T \)
7 \( 1 + (0.354 + 0.935i)T \)
11 \( 1 + (-0.568 - 0.822i)T \)
13 \( 1 + (-0.748 - 0.663i)T \)
17 \( 1 + (-0.885 + 0.464i)T \)
19 \( 1 + (0.663 - 0.748i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.568 + 0.822i)T \)
31 \( 1 + (0.822 + 0.568i)T \)
37 \( 1 + (-0.120 + 0.992i)T \)
41 \( 1 + (-0.822 + 0.568i)T \)
43 \( 1 + (-0.120 - 0.992i)T \)
47 \( 1 + (-0.970 + 0.239i)T \)
59 \( 1 + (0.970 - 0.239i)T \)
61 \( 1 + (0.464 - 0.885i)T \)
67 \( 1 + (0.663 + 0.748i)T \)
71 \( 1 + (0.992 - 0.120i)T \)
73 \( 1 + (0.464 + 0.885i)T \)
79 \( 1 + (-0.935 - 0.354i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (-0.970 - 0.239i)T \)
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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

−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