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Properties

Label 1-283-283.47-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.241 + 0.970i$
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  + (−0.231 + 0.972i)2-s + (0.911 − 0.410i)3-s + (−0.892 − 0.451i)4-s + (−0.317 + 0.948i)5-s + (0.188 + 0.982i)6-s + (−0.122 + 0.992i)7-s + (0.645 − 0.763i)8-s + (0.662 − 0.749i)9-s + (−0.848 − 0.528i)10-s + (0.0556 − 0.998i)11-s + (−0.999 − 0.0445i)12-s + (0.756 − 0.654i)13-s + (−0.937 − 0.348i)14-s + (0.100 + 0.994i)15-s + (0.593 + 0.805i)16-s + (0.770 + 0.637i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.972i)2-s + (0.911 − 0.410i)3-s + (−0.892 − 0.451i)4-s + (−0.317 + 0.948i)5-s + (0.188 + 0.982i)6-s + (−0.122 + 0.992i)7-s + (0.645 − 0.763i)8-s + (0.662 − 0.749i)9-s + (−0.848 − 0.528i)10-s + (0.0556 − 0.998i)11-s + (−0.999 − 0.0445i)12-s + (0.756 − 0.654i)13-s + (−0.937 − 0.348i)14-s + (0.100 + 0.994i)15-s + (0.593 + 0.805i)16-s + (0.770 + 0.637i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.241 + 0.970i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.241 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329964638 + 1.700793644i\)
\(L(\frac12)\) \(\approx\) \(1.329964638 + 1.700793644i\)
\(L(1)\) \(\approx\) \(1.078123904 + 0.6477890395i\)
\(L(1)\) \(\approx\) \(1.078123904 + 0.6477890395i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.231 + 0.972i)T \)
3 \( 1 + (0.911 - 0.410i)T \)
5 \( 1 + (-0.317 + 0.948i)T \)
7 \( 1 + (-0.122 + 0.992i)T \)
11 \( 1 + (0.0556 - 0.998i)T \)
13 \( 1 + (0.756 - 0.654i)T \)
17 \( 1 + (0.770 + 0.637i)T \)
19 \( 1 + (0.944 + 0.328i)T \)
23 \( 1 + (-0.0779 + 0.996i)T \)
29 \( 1 + (-0.420 - 0.907i)T \)
31 \( 1 + (0.253 + 0.967i)T \)
37 \( 1 + (0.987 + 0.155i)T \)
41 \( 1 + (-0.338 + 0.940i)T \)
43 \( 1 + (0.741 - 0.670i)T \)
47 \( 1 + (-0.188 + 0.982i)T \)
53 \( 1 + (-0.593 + 0.805i)T \)
59 \( 1 + (0.274 + 0.961i)T \)
61 \( 1 + (-0.0334 - 0.999i)T \)
67 \( 1 + (0.538 + 0.842i)T \)
71 \( 1 + (-0.892 + 0.451i)T \)
73 \( 1 + (-0.253 + 0.967i)T \)
79 \( 1 + (0.741 + 0.670i)T \)
83 \( 1 + (0.975 - 0.220i)T \)
89 \( 1 + (-0.210 - 0.977i)T \)
97 \( 1 + (0.902 - 0.431i)T \)
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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

−25.40066944211170083311339581069, −24.123860304484874375965533484070, −23.1518867762272044509814847682, −22.2230164039005470866885405311, −20.806619475145616050953748492547, −20.62817727169729053097793888038, −19.95242807101271519415884344082, −19.03638544876156721324291643360, −17.98714152502491699498013284663, −16.67703542273993447270986502845, −16.165606435100095799357231985362, −14.646802586947832999840672890510, −13.70727698895964889263910212743, −13.01989725736268019809205481020, −11.98825003536745165334898888836, −10.8428122046000049204339178267, −9.73015307920876788177496627344, −9.23203985843140645514363329496, −8.09387470596425181197497417397, −7.29378626101176497414377826636, −4.948985580285619939346991455274, −4.22064041567058346763201721578, −3.37184059944748959008668376653, −1.89382454807922919539019875338, −0.77676458672953328997439811895, 1.187951397653346645692273164112, 2.97038337133412939608052417906, 3.73098477576452939864794433719, 5.69630730430879683337913399000, 6.31141576949915660191955980260, 7.66752633745592563728416797348, 8.15732636039803075691452119516, 9.20028773462657854159115778062, 10.188797084729167120449926414355, 11.58654745197025624113727878018, 12.90677937299037976799327235252, 13.87293290999090343908226615776, 14.58679906041344806561078905490, 15.477262427041885320734908184269, 15.99936750912015919803266447250, 17.60204255529800482903569801276, 18.53823346415139442271267528595, 18.869380343225986334487015324402, 19.746181151921514835253759941688, 21.27327947671479056653962918727, 22.11100757052457411079613944196, 23.16907094292667890860879737798, 23.96878002792172938367057379052, 25.027299069374327417482502812340, 25.45669381682338248541316230632

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