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Properties

Label 1-283-283.188-r1-0-0
Degree $1$
Conductor $283$
Sign $0.608 + 0.793i$
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  + (−0.860 − 0.509i)2-s + (0.929 + 0.369i)3-s + (0.480 + 0.876i)4-s + (0.958 + 0.285i)5-s + (−0.610 − 0.791i)6-s + (0.0556 − 0.998i)7-s + (0.0334 − 0.999i)8-s + (0.726 + 0.687i)9-s + (−0.679 − 0.734i)10-s + (0.519 + 0.854i)11-s + (0.122 + 0.992i)12-s + (−0.380 + 0.924i)13-s + (−0.556 + 0.830i)14-s + (0.784 + 0.619i)15-s + (−0.538 + 0.842i)16-s + (−0.441 + 0.897i)17-s + ⋯
L(s)  = 1  + (−0.860 − 0.509i)2-s + (0.929 + 0.369i)3-s + (0.480 + 0.876i)4-s + (0.958 + 0.285i)5-s + (−0.610 − 0.791i)6-s + (0.0556 − 0.998i)7-s + (0.0334 − 0.999i)8-s + (0.726 + 0.687i)9-s + (−0.679 − 0.734i)10-s + (0.519 + 0.854i)11-s + (0.122 + 0.992i)12-s + (−0.380 + 0.924i)13-s + (−0.556 + 0.830i)14-s + (0.784 + 0.619i)15-s + (−0.538 + 0.842i)16-s + (−0.441 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.608 + 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.896069883 + 0.9350241111i\)
\(L(\frac12)\) \(\approx\) \(1.896069883 + 0.9350241111i\)
\(L(1)\) \(\approx\) \(1.207300055 + 0.1438787446i\)
\(L(1)\) \(\approx\) \(1.207300055 + 0.1438787446i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.860 - 0.509i)T \)
3 \( 1 + (0.929 + 0.369i)T \)
5 \( 1 + (0.958 + 0.285i)T \)
7 \( 1 + (0.0556 - 0.998i)T \)
11 \( 1 + (0.519 + 0.854i)T \)
13 \( 1 + (-0.380 + 0.924i)T \)
17 \( 1 + (-0.441 + 0.897i)T \)
19 \( 1 + (-0.991 + 0.133i)T \)
23 \( 1 + (0.984 + 0.177i)T \)
29 \( 1 + (0.695 + 0.718i)T \)
31 \( 1 + (-0.951 - 0.306i)T \)
37 \( 1 + (-0.937 - 0.348i)T \)
41 \( 1 + (-0.848 + 0.528i)T \)
43 \( 1 + (0.944 - 0.328i)T \)
47 \( 1 + (0.610 - 0.791i)T \)
53 \( 1 + (0.538 + 0.842i)T \)
59 \( 1 + (0.400 + 0.916i)T \)
61 \( 1 + (-0.296 - 0.955i)T \)
67 \( 1 + (-0.920 + 0.390i)T \)
71 \( 1 + (0.480 - 0.876i)T \)
73 \( 1 + (0.951 - 0.306i)T \)
79 \( 1 + (0.944 + 0.328i)T \)
83 \( 1 + (-0.575 + 0.818i)T \)
89 \( 1 + (0.188 - 0.982i)T \)
97 \( 1 + (-0.338 - 0.940i)T \)
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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

−25.22651932589874399601150590037, −24.70613360461576419160499770513, −24.05308351892625144544341497841, −22.47470277434322673405579139367, −21.31765379193532780545861290779, −20.57321320140129686397630181490, −19.49153147874695109155146528327, −18.81576979773468596366341276244, −17.93072373825506347019588921185, −17.19805690209260481195760119791, −15.96061600897547664273557593030, −15.07225638131092543293572906946, −14.273269457355555941043670910708, −13.31739800096518815902910910817, −12.21224315329142927533872101843, −10.793971564827948648339587575044, −9.62570328819110977514681185361, −8.88179972706470194849803524230, −8.376837570123362982578756494222, −6.99342016034429147405092098583, −6.083404090388560705987738919437, −5.045443758964081608850361421986, −2.8776318037670204403492231362, −2.0618922686567527345151465610, −0.75340915482983814920230534084, 1.53035405068044012027822820303, 2.207374897756577774755802999617, 3.60497825023102903991751126242, 4.53750341535259703956313613717, 6.71233615486382961562535043619, 7.28574374182216919598239211036, 8.71240641819235307517648245825, 9.360790117033315489678489720659, 10.321857375815563743640665037719, 10.83331544826243461784445211811, 12.49047571117632928322938311061, 13.383910171341272831120749541196, 14.36070619193865300102243972643, 15.241478568695160257964032892440, 16.79290053790318291130535006476, 17.10556402529624485691218009473, 18.26696837876111826413154061720, 19.35576840784309285831089280876, 19.9084685650024504972278892345, 20.910319552417475531727418182989, 21.4712790448655637596068492599, 22.36807058531242350774778252836, 23.90703480662993358980955474010, 25.11772473136205863462427320568, 25.69387638684833613599075255055

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