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Properties

Label 1-283-283.131-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.232 - 0.972i$
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  + (0.538 + 0.842i)2-s + (0.892 − 0.451i)3-s + (−0.420 + 0.907i)4-s + (−0.593 − 0.805i)5-s + (0.860 + 0.509i)6-s + (−0.296 + 0.955i)7-s + (−0.991 + 0.133i)8-s + (0.593 − 0.805i)9-s + (0.359 − 0.933i)10-s + (−0.420 − 0.907i)11-s + (0.0334 + 0.999i)12-s + (0.860 + 0.509i)13-s + (−0.964 + 0.264i)14-s + (−0.892 − 0.451i)15-s + (−0.645 − 0.763i)16-s + (−0.964 − 0.264i)17-s + ⋯
L(s)  = 1  + (0.538 + 0.842i)2-s + (0.892 − 0.451i)3-s + (−0.420 + 0.907i)4-s + (−0.593 − 0.805i)5-s + (0.860 + 0.509i)6-s + (−0.296 + 0.955i)7-s + (−0.991 + 0.133i)8-s + (0.593 − 0.805i)9-s + (0.359 − 0.933i)10-s + (−0.420 − 0.907i)11-s + (0.0334 + 0.999i)12-s + (0.860 + 0.509i)13-s + (−0.964 + 0.264i)14-s + (−0.892 − 0.451i)15-s + (−0.645 − 0.763i)16-s + (−0.964 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.232 - 0.972i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.232 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5769213044 - 0.7307169263i\)
\(L(\frac12)\) \(\approx\) \(0.5769213044 - 0.7307169263i\)
\(L(1)\) \(\approx\) \(1.180225293 + 0.1743272796i\)
\(L(1)\) \(\approx\) \(1.180225293 + 0.1743272796i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.538 + 0.842i)T \)
3 \( 1 + (0.892 - 0.451i)T \)
5 \( 1 + (-0.593 - 0.805i)T \)
7 \( 1 + (-0.296 + 0.955i)T \)
11 \( 1 + (-0.420 - 0.907i)T \)
13 \( 1 + (0.860 + 0.509i)T \)
17 \( 1 + (-0.964 - 0.264i)T \)
19 \( 1 + (-0.860 - 0.509i)T \)
23 \( 1 + (-0.944 - 0.328i)T \)
29 \( 1 + (-0.997 + 0.0667i)T \)
31 \( 1 + (0.979 - 0.199i)T \)
37 \( 1 + (-0.784 - 0.619i)T \)
41 \( 1 + (0.991 + 0.133i)T \)
43 \( 1 + (-0.231 - 0.972i)T \)
47 \( 1 + (-0.860 + 0.509i)T \)
53 \( 1 + (0.645 - 0.763i)T \)
59 \( 1 + (-0.824 - 0.565i)T \)
61 \( 1 + (0.359 + 0.933i)T \)
67 \( 1 + (0.0334 - 0.999i)T \)
71 \( 1 + (-0.420 - 0.907i)T \)
73 \( 1 + (-0.979 - 0.199i)T \)
79 \( 1 + (-0.231 + 0.972i)T \)
83 \( 1 + (-0.166 + 0.986i)T \)
89 \( 1 + (0.231 + 0.972i)T \)
97 \( 1 + (-0.944 - 0.328i)T \)
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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

−26.11163250659771197292294241089, −24.61096996866092403663408104297, −23.382550151157529154709921176720, −22.89072703313325232449740544471, −21.97060998407388749233595223971, −20.879548193840596936401386223508, −20.1977765639651970679253109424, −19.57180640862955685064499925824, −18.69710745192113274393763825555, −17.6746367189868294976089669890, −15.9233097109508472371173514622, −15.24752236382570061892879550596, −14.434631318696147377802505318757, −13.49298034377599474384500918641, −12.79541711502350061459355623867, −11.339011340616464919839528102509, −10.409102425856772324048245778578, −10.02243279834766469495146567947, −8.56249482028653053244517804577, −7.47750934498717112566031039547, −6.25436021927695253081148069636, −4.4711529393290703398152003344, −3.89712355350786422559981645415, −2.952814878764773745066138634866, −1.782284529439498819496747265697, 0.19438934093013476803896570877, 2.23857053331996009367560788298, 3.48664728827320185206348515672, 4.43181475712007890819031588615, 5.80077494265871926867074634290, 6.71826208379864642981902829992, 8.03361493224177679611789293346, 8.65672703007612322206169454043, 9.19990472889512940117328769171, 11.378879790699527125328216496415, 12.38649782384241319541217073303, 13.164104935795138428798940973900, 13.82612962652218831228233300506, 15.09237781015019443647442881269, 15.74518878631845636415610762691, 16.386771739534874103332221208105, 17.815421023963972371879758265839, 18.72313108519372857320669141938, 19.53955624243995865747314216074, 20.81538683771401092447760904112, 21.35839046159019797808299183472, 22.529350349773181160439956310936, 23.72985035269343274348405886821, 24.26906559568612591147243639259, 24.84190493137027067061148410039

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