Properties

Label 1-283-283.20-r1-0-0
Degree $1$
Conductor $283$
Sign $0.751 + 0.659i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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  + (−0.480 − 0.876i)2-s + (0.958 − 0.285i)3-s + (−0.538 + 0.842i)4-s + (−0.0556 + 0.998i)5-s + (−0.711 − 0.703i)6-s + (0.400 + 0.916i)7-s + (0.997 + 0.0667i)8-s + (0.836 − 0.547i)9-s + (0.902 − 0.431i)10-s + (0.999 − 0.0445i)11-s + (−0.274 + 0.961i)12-s + (−0.253 + 0.967i)13-s + (0.610 − 0.791i)14-s + (0.231 + 0.972i)15-s + (−0.420 − 0.907i)16-s + (0.380 − 0.924i)17-s + ⋯
L(s)  = 1  + (−0.480 − 0.876i)2-s + (0.958 − 0.285i)3-s + (−0.538 + 0.842i)4-s + (−0.0556 + 0.998i)5-s + (−0.711 − 0.703i)6-s + (0.400 + 0.916i)7-s + (0.997 + 0.0667i)8-s + (0.836 − 0.547i)9-s + (0.902 − 0.431i)10-s + (0.999 − 0.0445i)11-s + (−0.274 + 0.961i)12-s + (−0.253 + 0.967i)13-s + (0.610 − 0.791i)14-s + (0.231 + 0.972i)15-s + (−0.420 − 0.907i)16-s + (0.380 − 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.887098560 + 0.7111236789i\)
\(L(\frac12)\) \(\approx\) \(1.887098560 + 0.7111236789i\)
\(L(1)\) \(\approx\) \(1.207397231 - 0.04274096727i\)
\(L(1)\) \(\approx\) \(1.207397231 - 0.04274096727i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.480 - 0.876i)T \)
3 \( 1 + (0.958 - 0.285i)T \)
5 \( 1 + (-0.0556 + 0.998i)T \)
7 \( 1 + (0.400 + 0.916i)T \)
11 \( 1 + (0.999 - 0.0445i)T \)
13 \( 1 + (-0.253 + 0.967i)T \)
17 \( 1 + (0.380 - 0.924i)T \)
19 \( 1 + (-0.964 + 0.264i)T \)
23 \( 1 + (-0.770 + 0.637i)T \)
29 \( 1 + (-0.0334 + 0.999i)T \)
31 \( 1 + (0.911 - 0.410i)T \)
37 \( 1 + (-0.188 + 0.982i)T \)
41 \( 1 + (0.556 + 0.830i)T \)
43 \( 1 + (-0.784 + 0.619i)T \)
47 \( 1 + (0.711 - 0.703i)T \)
53 \( 1 + (0.420 - 0.907i)T \)
59 \( 1 + (0.975 + 0.220i)T \)
61 \( 1 + (-0.824 + 0.565i)T \)
67 \( 1 + (-0.695 + 0.718i)T \)
71 \( 1 + (-0.538 - 0.842i)T \)
73 \( 1 + (-0.911 - 0.410i)T \)
79 \( 1 + (-0.784 - 0.619i)T \)
83 \( 1 + (0.984 + 0.177i)T \)
89 \( 1 + (0.144 + 0.989i)T \)
97 \( 1 + (0.937 + 0.348i)T \)
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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

−25.17528198341562838428057521383, −24.57587025527188640741543144578, −23.832639042023806827591185427550, −22.78394904494704272093240224700, −21.49014833406312073671674232644, −20.40463678576120509201866930807, −19.774310082774052015545648125664, −19.09660489789015038379714119167, −17.47430606368334420758818954559, −17.077772789534356834646708816821, −16.036392428758171632997813960667, −15.11236919097441225620115814801, −14.29866206303644059822493641888, −13.47795438788000349223428816039, −12.45946764279865986450060751003, −10.615067379452483526649729384214, −9.893173522391522698677912866810, −8.760902101102074729030779963808, −8.191429074035149597354694041967, −7.310566400706335156164668326933, −5.935724229056268457845550585520, −4.524874849559648966453382283398, −3.98557191865395856840560325884, −1.82112643943504457846355331994, −0.66455790941368785480162319899, 1.54458286262771555446334703656, 2.403259224521546546765760166193, 3.35942176188052279241876535744, 4.44390061597911879589233968402, 6.4493930084746905461847172843, 7.4741747646220911259775531800, 8.5159735258517627839835791795, 9.33139604342760740989347421892, 10.170711044559442145950735345290, 11.6759356540488948724759879848, 11.937977460369264682447477969168, 13.41486042925549381197326071983, 14.311512827372876309500940953519, 14.93672857158101221210894669696, 16.33944258829414184431537377296, 17.704348721551250055387152757768, 18.47181234460078902035325175785, 19.1115874544563218075890619000, 19.73669741921928832128339328260, 20.94324535094260667128722798752, 21.64576857408709831814748894432, 22.34426922906669661375102450288, 23.64468843191098657999957742546, 24.94874815463638373316547369564, 25.583920942859574345578232410305

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