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Properties

Label 1-283-283.40-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.635 - 0.772i$
Analytic cond. $1.31424$
Root an. cond. $1.31424$
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  + (−0.420 + 0.907i)2-s + (−0.993 − 0.111i)3-s + (−0.645 − 0.763i)4-s + (−0.679 − 0.734i)5-s + (0.519 − 0.854i)6-s + (0.902 − 0.431i)7-s + (0.964 − 0.264i)8-s + (0.975 + 0.220i)9-s + (0.951 − 0.306i)10-s + (−0.338 − 0.940i)11-s + (0.556 + 0.830i)12-s + (−0.999 − 0.0222i)13-s + (0.0111 + 0.999i)14-s + (0.593 + 0.805i)15-s + (−0.166 + 0.986i)16-s + (−0.871 + 0.490i)17-s + ⋯
L(s)  = 1  + (−0.420 + 0.907i)2-s + (−0.993 − 0.111i)3-s + (−0.645 − 0.763i)4-s + (−0.679 − 0.734i)5-s + (0.519 − 0.854i)6-s + (0.902 − 0.431i)7-s + (0.964 − 0.264i)8-s + (0.975 + 0.220i)9-s + (0.951 − 0.306i)10-s + (−0.338 − 0.940i)11-s + (0.556 + 0.830i)12-s + (−0.999 − 0.0222i)13-s + (0.0111 + 0.999i)14-s + (0.593 + 0.805i)15-s + (−0.166 + 0.986i)16-s + (−0.871 + 0.490i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.635 - 0.772i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.635 - 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09172811325 - 0.1942194624i\)
\(L(\frac12)\) \(\approx\) \(0.09172811325 - 0.1942194624i\)
\(L(1)\) \(\approx\) \(0.4597536053 + 0.007775978203i\)
\(L(1)\) \(\approx\) \(0.4597536053 + 0.007775978203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.420 + 0.907i)T \)
3 \( 1 + (-0.993 - 0.111i)T \)
5 \( 1 + (-0.679 - 0.734i)T \)
7 \( 1 + (0.902 - 0.431i)T \)
11 \( 1 + (-0.338 - 0.940i)T \)
13 \( 1 + (-0.999 - 0.0222i)T \)
17 \( 1 + (-0.871 + 0.490i)T \)
19 \( 1 + (0.480 + 0.876i)T \)
23 \( 1 + (0.144 - 0.989i)T \)
29 \( 1 + (0.991 - 0.133i)T \)
31 \( 1 + (-0.798 - 0.602i)T \)
37 \( 1 + (-0.958 - 0.285i)T \)
41 \( 1 + (-0.253 - 0.967i)T \)
43 \( 1 + (-0.892 + 0.451i)T \)
47 \( 1 + (0.519 + 0.854i)T \)
53 \( 1 + (-0.166 - 0.986i)T \)
59 \( 1 + (-0.987 - 0.155i)T \)
61 \( 1 + (-0.741 + 0.670i)T \)
67 \( 1 + (-0.997 - 0.0667i)T \)
71 \( 1 + (-0.645 + 0.763i)T \)
73 \( 1 + (-0.798 + 0.602i)T \)
79 \( 1 + (-0.892 - 0.451i)T \)
83 \( 1 + (0.188 + 0.982i)T \)
89 \( 1 + (0.0556 - 0.998i)T \)
97 \( 1 + (-0.929 + 0.369i)T \)
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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

−26.41351719005997024409660350439, −25.01670472298652317891572039915, −23.75753935978085533762891551532, −23.05302574267453075287323887817, −21.96978781092585195148379243385, −21.72620360799293021696097428597, −20.36047921708223151803588115771, −19.56021785593997490337074972982, −18.34383947179348565189749360250, −17.91435798541018997628550297925, −17.17921968853262616715138019946, −15.76491076119930999831281811150, −15.01423355294171184167690939533, −13.61784644864545406613838046519, −12.23535495887442302704525241043, −11.80896157935567759844515631589, −10.965486424392523383994361798662, −10.17984462280349106883161437708, −9.06059842225132747823480942144, −7.597701723660443088097851698555, −7.009285467171037640203252809646, −5.03365891642633027541666827162, −4.527583982022053213272540285459, −2.95511898987899210064560596053, −1.72689719184428743385227857057, 0.19913211101432577855653461043, 1.4908462365297103901335260789, 4.18166610946410590558173718652, 4.92271859788924118430652916116, 5.80566094301849513125789977126, 7.08662077421836288390197462284, 7.91636777350884825450345821633, 8.76040070815831501741964935221, 10.2237898655004699296262147980, 11.0258866297188810169790482039, 12.09706165197427119513981356928, 13.1610791650086804868212667875, 14.29885993581277830386397163986, 15.40158718901296477732764378738, 16.29465236597762353282142635556, 16.9184516269837001567659828132, 17.651664995773901751158025961827, 18.64062838471978004862945832401, 19.52532888217527723603798176541, 20.69354711437786337927009892358, 21.910879630550703255151282190237, 22.86469788129627898850168908830, 23.780482716818298971372491756741, 24.314367032116078300122386516788, 24.72930656354329916139390435300

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