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Properties

Label 1-283-283.125-r1-0-0
Degree $1$
Conductor $283$
Sign $0.636 + 0.770i$
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  + (0.741 + 0.670i)2-s + (−0.991 + 0.133i)3-s + (0.100 + 0.994i)4-s + (−0.964 − 0.264i)5-s + (−0.824 − 0.565i)6-s + (0.860 + 0.509i)7-s + (−0.593 + 0.805i)8-s + (0.964 − 0.264i)9-s + (−0.538 − 0.842i)10-s + (0.100 − 0.994i)11-s + (−0.231 − 0.972i)12-s + (−0.824 − 0.565i)13-s + (0.296 + 0.955i)14-s + (0.991 + 0.133i)15-s + (−0.979 + 0.199i)16-s + (0.296 − 0.955i)17-s + ⋯
L(s)  = 1  + (0.741 + 0.670i)2-s + (−0.991 + 0.133i)3-s + (0.100 + 0.994i)4-s + (−0.964 − 0.264i)5-s + (−0.824 − 0.565i)6-s + (0.860 + 0.509i)7-s + (−0.593 + 0.805i)8-s + (0.964 − 0.264i)9-s + (−0.538 − 0.842i)10-s + (0.100 − 0.994i)11-s + (−0.231 − 0.972i)12-s + (−0.824 − 0.565i)13-s + (0.296 + 0.955i)14-s + (0.991 + 0.133i)15-s + (−0.979 + 0.199i)16-s + (0.296 − 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.636 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.636 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.636 + 0.770i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.636 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.602704270 + 0.7548760716i\)
\(L(\frac12)\) \(\approx\) \(1.602704270 + 0.7548760716i\)
\(L(1)\) \(\approx\) \(1.039778823 + 0.4249088552i\)
\(L(1)\) \(\approx\) \(1.039778823 + 0.4249088552i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.741 + 0.670i)T \)
3 \( 1 + (-0.991 + 0.133i)T \)
5 \( 1 + (-0.964 - 0.264i)T \)
7 \( 1 + (0.860 + 0.509i)T \)
11 \( 1 + (0.100 - 0.994i)T \)
13 \( 1 + (-0.824 - 0.565i)T \)
17 \( 1 + (0.296 - 0.955i)T \)
19 \( 1 + (0.824 + 0.565i)T \)
23 \( 1 + (0.695 - 0.718i)T \)
29 \( 1 + (-0.892 + 0.451i)T \)
31 \( 1 + (0.166 - 0.986i)T \)
37 \( 1 + (0.0334 + 0.999i)T \)
41 \( 1 + (0.593 + 0.805i)T \)
43 \( 1 + (0.997 - 0.0667i)T \)
47 \( 1 + (0.824 - 0.565i)T \)
53 \( 1 + (0.979 + 0.199i)T \)
59 \( 1 + (0.480 + 0.876i)T \)
61 \( 1 + (-0.538 + 0.842i)T \)
67 \( 1 + (-0.231 + 0.972i)T \)
71 \( 1 + (0.100 - 0.994i)T \)
73 \( 1 + (-0.166 - 0.986i)T \)
79 \( 1 + (0.997 + 0.0667i)T \)
83 \( 1 + (0.920 - 0.390i)T \)
89 \( 1 + (-0.997 + 0.0667i)T \)
97 \( 1 + (0.695 - 0.718i)T \)
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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

−24.63818756648081604451431160769, −23.96474217398161178130551586908, −23.32082934827012766971886543886, −22.64498911399943181353796023422, −21.74352756384416920986199542204, −20.839198495768183984284891469819, −19.75726607297655474718483614155, −19.07957096521172494295169250704, −17.92720971063502153166935106971, −17.10872723047332509540608472085, −15.78499198429901477287976403197, −14.98266159881460732319549614104, −14.08988217116925724415664041632, −12.70998130201395065762343664676, −12.07384167424262117713169434824, −11.24135112674385518427267486401, −10.630380700819208764125513987039, −9.46456720942683199916010949738, −7.49978551988636359349911298247, −6.97499078858958929269637816847, −5.43208774374815676942699377444, −4.577364801735877972712376234109, −3.82612356625848148103794892599, −2.04810782757673517142254867176, −0.83760711383807589892871368963, 0.72209126143242919363061558349, 2.9348207896852940692039327096, 4.23636384029436137386432003054, 5.13265741845768114961261728987, 5.78202221004721641197151280464, 7.237013012172005486457898481983, 7.90042033682965297047104255586, 9.10498281056980398475515387459, 10.861047818908727122707597513820, 11.7660855061893401829924460700, 12.13883836447532661340793325346, 13.32640650333478323875282386858, 14.66880694586370845067660139712, 15.31755475030313503691233424059, 16.37707021980005801723606622314, 16.792525854366911134888307724730, 18.02048545323386886841836332602, 18.80885575192801793627750287678, 20.43459432607438385237043848562, 21.13736250024301992645762830172, 22.32125885244030299872632389284, 22.64104519863182589438733154753, 23.8216125160539051624069410299, 24.37023064890683608204153727525, 24.918608196211246345504666892

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