↑

Properties

Label 1-283-283.181-r0-0-0
Degree $1$
Conductor $283$
Sign $0.165 + 0.986i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.420 + 0.907i)2-s + (0.593 − 0.805i)3-s + (−0.645 − 0.763i)4-s + (−0.296 + 0.955i)5-s + (0.480 + 0.876i)6-s + (−0.824 − 0.565i)7-s + (0.964 − 0.264i)8-s + (−0.296 − 0.955i)9-s + (−0.741 − 0.670i)10-s + (−0.645 + 0.763i)11-s + (−0.997 + 0.0667i)12-s + (0.480 + 0.876i)13-s + (0.860 − 0.509i)14-s + (0.593 + 0.805i)15-s + (−0.166 + 0.986i)16-s + (0.860 + 0.509i)17-s + ⋯
L(s)  = 1  + (−0.420 + 0.907i)2-s + (0.593 − 0.805i)3-s + (−0.645 − 0.763i)4-s + (−0.296 + 0.955i)5-s + (0.480 + 0.876i)6-s + (−0.824 − 0.565i)7-s + (0.964 − 0.264i)8-s + (−0.296 − 0.955i)9-s + (−0.741 − 0.670i)10-s + (−0.645 + 0.763i)11-s + (−0.997 + 0.0667i)12-s + (0.480 + 0.876i)13-s + (0.860 − 0.509i)14-s + (0.593 + 0.805i)15-s + (−0.166 + 0.986i)16-s + (0.860 + 0.509i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.165 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7058817969 + 0.5971959974i\)
\(L(\frac12)\) \(\approx\) \(0.7058817969 + 0.5971959974i\)
\(L(1)\) \(\approx\) \(0.8103788381 + 0.3147976737i\)
\(L(1)\) \(\approx\) \(0.8103788381 + 0.3147976737i\)

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 \( 1 + (0.593 - 0.805i)T \)
5 \( 1 + (-0.296 + 0.955i)T \)
7 \( 1 + (-0.824 - 0.565i)T \)
11 \( 1 + (-0.645 + 0.763i)T \)
13 \( 1 + (0.480 + 0.876i)T \)
17 \( 1 + (0.860 + 0.509i)T \)
19 \( 1 + (0.480 + 0.876i)T \)
23 \( 1 + (0.784 + 0.619i)T \)
29 \( 1 + (0.991 - 0.133i)T \)
31 \( 1 + (0.920 - 0.390i)T \)
37 \( 1 + (0.231 + 0.972i)T \)
41 \( 1 + (0.964 + 0.264i)T \)
43 \( 1 + (-0.892 + 0.451i)T \)
47 \( 1 + (0.480 - 0.876i)T \)
53 \( 1 + (-0.166 - 0.986i)T \)
59 \( 1 + (0.359 + 0.933i)T \)
61 \( 1 + (-0.741 + 0.670i)T \)
67 \( 1 + (-0.997 - 0.0667i)T \)
71 \( 1 + (-0.645 + 0.763i)T \)
73 \( 1 + (0.920 + 0.390i)T \)
79 \( 1 + (-0.892 - 0.451i)T \)
83 \( 1 + (-0.944 - 0.328i)T \)
89 \( 1 + (-0.892 + 0.451i)T \)
97 \( 1 + (0.784 + 0.619i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.47981193953130261770178630659, −24.8815750115033588468923582249, −23.267855575718658621507835028027, −22.41764326191971137124427185497, −21.353517756234647845829275448064, −20.88497696455013777653071007423, −19.93587901296233178833629852690, −19.30576024476322279999257129133, −18.35978047678232380061163324714, −17.033261630548810497416439546173, −16.04904953285528787394653008637, −15.65597032044136549797529288913, −13.951851360735616009852738156155, −13.11193976533105197120699711881, −12.31096336703666904028583031668, −11.1027473254523241118664969303, −10.17281206767315399931975935056, −9.19944514839835199298537255570, −8.60634879470248777167355017348, −7.72180166946532607013396138488, −5.54037575893121971000267728886, −4.6367698792919315726432159792, −3.26536880651865163470476640935, −2.76745295425738705799892114657, −0.76065818904809322227364241329, 1.34909148844679514375794180312, 2.97128505795648332448815363713, 4.12277762011368612764703709799, 5.96009566726531044343620928130, 6.78812245268253418149035289592, 7.4612328076014995587636560432, 8.2858252281241105971132872335, 9.67208661733307533502412599958, 10.28726039384926509244410173213, 11.81941525764419383329795912503, 13.129873075695106967186347239914, 13.871257168899311307048836008851, 14.71935771826910821780780060256, 15.53846393563502690333957261425, 16.600144453112688961004258115503, 17.69044984176957541194535493119, 18.59244288748368824478669419655, 19.099381349867985669873537090718, 19.88380387611972119751598295546, 21.18481622646060596981382963614, 22.865368335972974135553999365694, 23.15762898398326886420398311158, 23.90541580761236567130654104112, 25.2319042105092678002035192201, 25.759867030955235751585223134707

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