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Properties

Label 1-283-283.79-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.845 - 0.533i$
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  + (0.979 − 0.199i)2-s + (−0.860 + 0.509i)3-s + (0.920 − 0.390i)4-s + (−0.480 − 0.876i)5-s + (−0.741 + 0.670i)6-s + (−0.538 + 0.842i)7-s + (0.824 − 0.565i)8-s + (0.480 − 0.876i)9-s + (−0.645 − 0.763i)10-s + (0.920 + 0.390i)11-s + (−0.593 + 0.805i)12-s + (−0.741 + 0.670i)13-s + (−0.359 + 0.933i)14-s + (0.860 + 0.509i)15-s + (0.695 − 0.718i)16-s + (−0.359 − 0.933i)17-s + ⋯
L(s)  = 1  + (0.979 − 0.199i)2-s + (−0.860 + 0.509i)3-s + (0.920 − 0.390i)4-s + (−0.480 − 0.876i)5-s + (−0.741 + 0.670i)6-s + (−0.538 + 0.842i)7-s + (0.824 − 0.565i)8-s + (0.480 − 0.876i)9-s + (−0.645 − 0.763i)10-s + (0.920 + 0.390i)11-s + (−0.593 + 0.805i)12-s + (−0.741 + 0.670i)13-s + (−0.359 + 0.933i)14-s + (0.860 + 0.509i)15-s + (0.695 − 0.718i)16-s + (−0.359 − 0.933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.845 - 0.533i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.845 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2582037411 - 0.8932114443i\)
\(L(\frac12)\) \(\approx\) \(0.2582037411 - 0.8932114443i\)
\(L(1)\) \(\approx\) \(1.086650870 - 0.2198973118i\)
\(L(1)\) \(\approx\) \(1.086650870 - 0.2198973118i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.979 - 0.199i)T \)
3 \( 1 + (-0.860 + 0.509i)T \)
5 \( 1 + (-0.480 - 0.876i)T \)
7 \( 1 + (-0.538 + 0.842i)T \)
11 \( 1 + (0.920 + 0.390i)T \)
13 \( 1 + (-0.741 + 0.670i)T \)
17 \( 1 + (-0.359 - 0.933i)T \)
19 \( 1 + (0.741 - 0.670i)T \)
23 \( 1 + (-0.997 + 0.0667i)T \)
29 \( 1 + (-0.296 - 0.955i)T \)
31 \( 1 + (-0.784 - 0.619i)T \)
37 \( 1 + (-0.991 + 0.133i)T \)
41 \( 1 + (-0.824 - 0.565i)T \)
43 \( 1 + (-0.964 + 0.264i)T \)
47 \( 1 + (0.741 + 0.670i)T \)
53 \( 1 + (-0.695 - 0.718i)T \)
59 \( 1 + (-0.420 - 0.907i)T \)
61 \( 1 + (-0.645 + 0.763i)T \)
67 \( 1 + (-0.593 - 0.805i)T \)
71 \( 1 + (0.920 + 0.390i)T \)
73 \( 1 + (0.784 - 0.619i)T \)
79 \( 1 + (-0.964 - 0.264i)T \)
83 \( 1 + (-0.0334 - 0.999i)T \)
89 \( 1 + (0.964 - 0.264i)T \)
97 \( 1 + (-0.997 + 0.0667i)T \)
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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

−25.49919088846096812629636915408, −24.507639761816095665307517235987, −23.71084558566013873509084182432, −23.01848475266052780286958063616, −22.12558924261437610639169921801, −21.96502568951863203733362197050, −20.02485511560327198951020301495, −19.6441273232943784228592662846, −18.39152670806097054242539380462, −17.20670246146188141600828392143, −16.54143310587489639266651129387, −15.56738272505785881710274577867, −14.4350802750538444926497063967, −13.70677741657284115207063036090, −12.54504360525327816604954804296, −11.91551409348355152782290203314, −10.83715547713144278599200745806, −10.23677497055910229770525662767, −7.98233060142055082635369348511, −7.0900396698052381389657092218, −6.48751263343855338392359302582, −5.472276914321354513013207162245, −4.06627278697081704142256402392, −3.26054888867370940051056711943, −1.611426129445530975915614536104, 0.214823619636846387221255932766, 1.88170712420969182783549548667, 3.517987554976295936466854409146, 4.53523903340348643748194683195, 5.22845826645377336352057278108, 6.30327221212683571017455771485, 7.289126124757714624654721708, 9.208705538162045552698059089479, 9.7701009287808749523299251625, 11.45110487659673437780623500056, 11.88167314775231200610287124296, 12.49535502923419761717456071547, 13.705109993527787917778217564668, 15.0206316789174230058052532068, 15.74651880923186449763668678123, 16.406936855956683294968125363992, 17.32363306565782819189706888910, 18.77973616934424380006706232118, 19.85759559695322031762069481835, 20.55611610131852946656268841203, 21.69374188031858318346543689503, 22.28181406796228431889916883284, 22.90689441579462997277679789098, 24.195313709238103488596810018703, 24.42656461476375134147301835528

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