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Properties

Label 1-283-283.161-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.381 - 0.924i$
Analytic cond. $1.31424$
Root an. cond. $1.31424$
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  + (0.695 + 0.718i)2-s + (−0.538 + 0.842i)3-s + (−0.0334 + 0.999i)4-s + (−0.420 + 0.907i)5-s + (−0.979 + 0.199i)6-s + (−0.645 − 0.763i)7-s + (−0.741 + 0.670i)8-s + (−0.420 − 0.907i)9-s + (−0.944 + 0.328i)10-s + (−0.0334 − 0.999i)11-s + (−0.824 − 0.565i)12-s + (−0.979 + 0.199i)13-s + (0.100 − 0.994i)14-s + (−0.538 − 0.842i)15-s + (−0.997 − 0.0667i)16-s + (0.100 + 0.994i)17-s + ⋯
L(s)  = 1  + (0.695 + 0.718i)2-s + (−0.538 + 0.842i)3-s + (−0.0334 + 0.999i)4-s + (−0.420 + 0.907i)5-s + (−0.979 + 0.199i)6-s + (−0.645 − 0.763i)7-s + (−0.741 + 0.670i)8-s + (−0.420 − 0.907i)9-s + (−0.944 + 0.328i)10-s + (−0.0334 − 0.999i)11-s + (−0.824 − 0.565i)12-s + (−0.979 + 0.199i)13-s + (0.100 − 0.994i)14-s + (−0.538 − 0.842i)15-s + (−0.997 − 0.0667i)16-s + (0.100 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.381 - 0.924i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.381 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2749776402 + 0.4111477345i\)
\(L(\frac12)\) \(\approx\) \(-0.2749776402 + 0.4111477345i\)
\(L(1)\) \(\approx\) \(0.5020173180 + 0.6264198274i\)
\(L(1)\) \(\approx\) \(0.5020173180 + 0.6264198274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.695 + 0.718i)T \)
3 \( 1 + (-0.538 + 0.842i)T \)
5 \( 1 + (-0.420 + 0.907i)T \)
7 \( 1 + (-0.645 - 0.763i)T \)
11 \( 1 + (-0.0334 - 0.999i)T \)
13 \( 1 + (-0.979 + 0.199i)T \)
17 \( 1 + (0.100 + 0.994i)T \)
19 \( 1 + (-0.979 + 0.199i)T \)
23 \( 1 + (0.964 + 0.264i)T \)
29 \( 1 + (0.359 + 0.933i)T \)
31 \( 1 + (-0.892 - 0.451i)T \)
37 \( 1 + (0.860 + 0.509i)T \)
41 \( 1 + (-0.741 - 0.670i)T \)
43 \( 1 + (0.480 + 0.876i)T \)
47 \( 1 + (-0.979 - 0.199i)T \)
53 \( 1 + (-0.997 + 0.0667i)T \)
59 \( 1 + (-0.166 + 0.986i)T \)
61 \( 1 + (-0.944 - 0.328i)T \)
67 \( 1 + (-0.824 + 0.565i)T \)
71 \( 1 + (-0.0334 - 0.999i)T \)
73 \( 1 + (-0.892 + 0.451i)T \)
79 \( 1 + (0.480 - 0.876i)T \)
83 \( 1 + (0.991 + 0.133i)T \)
89 \( 1 + (0.480 + 0.876i)T \)
97 \( 1 + (0.964 + 0.264i)T \)
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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

−24.959793757388657337039665804804, −23.82584406740385437157260848743, −23.08745421063322821507704931917, −22.44630626173050465716294089831, −21.42539873948386936987498914177, −20.29082509126052667933959127507, −19.54840967894852659024640563089, −18.86460156434509663212339291023, −17.79730274387917718485658557457, −16.706839729758194232906518354739, −15.60261874870920797799650731635, −14.681258957600281338340875419723, −13.263809865295609751857584538518, −12.61826301858971202518063480895, −12.170596555008221533980963676918, −11.260697897253116480704967698862, −9.88467221497024842250261110042, −8.952331034514552050011095146215, −7.4643259209550613843777054793, −6.396879526894169327502050661706, −5.20535646913725573560864355370, −4.60975150196056052048592409454, −2.84007814030629029584033331121, −1.85542244935573115940086979245, −0.264950753198858125567169302722, 3.02017019233999039025755064717, 3.71323463455339460946849207302, 4.69478226726303724838527120629, 6.04054749901502754509976751962, 6.68098454112020529740802998269, 7.7886922695219428636067519997, 9.11040038723053084540682935939, 10.449738696041613689803604179597, 11.113618436962372365766193156119, 12.24949055842391153604224981687, 13.3159120316082901295526260220, 14.60708444974997249044461763936, 14.948866070450021013774355939950, 16.154637305901373096753379912707, 16.7425455526360542014412636949, 17.52015945656785141598281059872, 18.925996905635139705518449071894, 19.92394597758250565162852412250, 21.29366525906135093978714113056, 21.897786754742506100954069863148, 22.58444017305336465237447659435, 23.498354946777571928672743358814, 23.91838846114229556306000680593, 25.53489519440456144058846466312, 26.22201838623592314514986207221

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