↑

Properties

Label 1-283-283.58-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.429 - 0.903i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.695 + 0.718i)2-s + (0.538 + 0.842i)3-s + (−0.0334 − 0.999i)4-s + (0.420 + 0.907i)5-s + (−0.979 − 0.199i)6-s + (−0.645 + 0.763i)7-s + (0.741 + 0.670i)8-s + (−0.420 + 0.907i)9-s + (−0.944 − 0.328i)10-s + (−0.0334 + 0.999i)11-s + (0.824 − 0.565i)12-s + (−0.979 − 0.199i)13-s + (−0.100 − 0.994i)14-s + (−0.538 + 0.842i)15-s + (−0.997 + 0.0667i)16-s + (−0.100 + 0.994i)17-s + ⋯
L(s)  = 1  + (−0.695 + 0.718i)2-s + (0.538 + 0.842i)3-s + (−0.0334 − 0.999i)4-s + (0.420 + 0.907i)5-s + (−0.979 − 0.199i)6-s + (−0.645 + 0.763i)7-s + (0.741 + 0.670i)8-s + (−0.420 + 0.907i)9-s + (−0.944 − 0.328i)10-s + (−0.0334 + 0.999i)11-s + (0.824 − 0.565i)12-s + (−0.979 − 0.199i)13-s + (−0.100 − 0.994i)14-s + (−0.538 + 0.842i)15-s + (−0.997 + 0.0667i)16-s + (−0.100 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.429 - 0.903i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.429 - 0.903i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6027707499 + 0.9538798421i\)
\(L(\frac12)\) \(\approx\) \(-0.6027707499 + 0.9538798421i\)
\(L(1)\) \(\approx\) \(0.4400152462 + 0.7188517138i\)
\(L(1)\) \(\approx\) \(0.4400152462 + 0.7188517138i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.695 + 0.718i)T \)
3 \( 1 + (0.538 + 0.842i)T \)
5 \( 1 + (0.420 + 0.907i)T \)
7 \( 1 + (-0.645 + 0.763i)T \)
11 \( 1 + (-0.0334 + 0.999i)T \)
13 \( 1 + (-0.979 - 0.199i)T \)
17 \( 1 + (-0.100 + 0.994i)T \)
19 \( 1 + (0.979 + 0.199i)T \)
23 \( 1 + (0.964 - 0.264i)T \)
29 \( 1 + (0.359 - 0.933i)T \)
31 \( 1 + (0.892 - 0.451i)T \)
37 \( 1 + (-0.860 + 0.509i)T \)
41 \( 1 + (-0.741 + 0.670i)T \)
43 \( 1 + (-0.480 + 0.876i)T \)
47 \( 1 + (0.979 - 0.199i)T \)
53 \( 1 + (0.997 + 0.0667i)T \)
59 \( 1 + (-0.166 - 0.986i)T \)
61 \( 1 + (-0.944 + 0.328i)T \)
67 \( 1 + (0.824 + 0.565i)T \)
71 \( 1 + (-0.0334 + 0.999i)T \)
73 \( 1 + (-0.892 - 0.451i)T \)
79 \( 1 + (-0.480 - 0.876i)T \)
83 \( 1 + (0.991 - 0.133i)T \)
89 \( 1 + (0.480 - 0.876i)T \)
97 \( 1 + (0.964 - 0.264i)T \)
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

−24.833355981195081044502064475015, −24.158497276375796957438227748357, −22.904653001350589550332376215324, −21.750925914529500937987859514514, −20.753702819559980450677527710949, −20.02631829525606128174414621622, −19.42473077763621591153839473841, −18.52763259196656122172771681896, −17.46295397015266412207828368827, −16.79742798096948683953108494916, −15.86290192718025025893326364012, −13.782861496959308139029100892463, −13.64533902060202154723576926790, −12.46807587945913207461961728450, −11.81077163556741257583210284886, −10.39898219482820730989467597618, −9.30375225666765775153966272987, −8.79634978084904900852919856683, −7.537820073963375047603943570327, −6.81142358836761366480718426451, −5.10453838912333480888755388285, −3.5190223146992203834598157000, −2.606109167418416195216839128318, −1.174846587807873413093595160849, −0.436342422898845606144600318275, 2.06277557937155880756626070319, 3.01501511907073385989258060370, 4.69561762232175967646359803469, 5.7500103094180832525702585779, 6.831385616681818143183132059173, 7.86192594899353163896640764127, 9.02651093784371890578695661866, 10.00135057363322891761009546030, 10.17453124078595970825867714137, 11.6788864409050497364319848654, 13.269005715807695337077322042820, 14.409797244718033927754607067, 15.17378839913311825409298281363, 15.50285580315283378492935433168, 16.85603377642277029949208756194, 17.58624186418008115562484216886, 18.754049882941785755112478316277, 19.378806712350284348355968818221, 20.340593619569320076182814799719, 21.587417979993199350787096615832, 22.44786550384200313260497655318, 23.03292780564319806082024681914, 24.81978096744031310453189404379, 25.10664621716583728287362396099, 26.26651083335681487559953066989

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