Properties

Label 1-283-283.30-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.815 - 0.579i$
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 + (−0.784 + 0.619i)3-s + (0.480 + 0.876i)4-s + (−0.231 − 0.972i)5-s + (0.991 − 0.133i)6-s + (−0.892 + 0.451i)7-s + (0.0334 − 0.999i)8-s + (0.231 − 0.972i)9-s + (−0.296 + 0.955i)10-s + (0.480 − 0.876i)11-s + (−0.920 − 0.390i)12-s + (0.991 − 0.133i)13-s + (0.997 + 0.0667i)14-s + (0.784 + 0.619i)15-s + (−0.538 + 0.842i)16-s + (0.997 − 0.0667i)17-s + ⋯
L(s)  = 1  + (−0.860 − 0.509i)2-s + (−0.784 + 0.619i)3-s + (0.480 + 0.876i)4-s + (−0.231 − 0.972i)5-s + (0.991 − 0.133i)6-s + (−0.892 + 0.451i)7-s + (0.0334 − 0.999i)8-s + (0.231 − 0.972i)9-s + (−0.296 + 0.955i)10-s + (0.480 − 0.876i)11-s + (−0.920 − 0.390i)12-s + (0.991 − 0.133i)13-s + (0.997 + 0.0667i)14-s + (0.784 + 0.619i)15-s + (−0.538 + 0.842i)16-s + (0.997 − 0.0667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.815 - 0.579i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.815 - 0.579i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1327913893 - 0.4163302889i\)
\(L(\frac12)\) \(\approx\) \(0.1327913893 - 0.4163302889i\)
\(L(1)\) \(\approx\) \(0.4792726177 - 0.1348869372i\)
\(L(1)\) \(\approx\) \(0.4792726177 - 0.1348869372i\)

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 \( 1 + (-0.784 + 0.619i)T \)
5 \( 1 + (-0.231 - 0.972i)T \)
7 \( 1 + (-0.892 + 0.451i)T \)
11 \( 1 + (0.480 - 0.876i)T \)
13 \( 1 + (0.991 - 0.133i)T \)
17 \( 1 + (0.997 - 0.0667i)T \)
19 \( 1 + (-0.991 + 0.133i)T \)
23 \( 1 + (-0.645 + 0.763i)T \)
29 \( 1 + (0.695 + 0.718i)T \)
31 \( 1 + (0.741 - 0.670i)T \)
37 \( 1 + (0.166 + 0.986i)T \)
41 \( 1 + (-0.0334 - 0.999i)T \)
43 \( 1 + (0.944 - 0.328i)T \)
47 \( 1 + (-0.991 - 0.133i)T \)
53 \( 1 + (0.538 + 0.842i)T \)
59 \( 1 + (0.593 - 0.805i)T \)
61 \( 1 + (-0.296 - 0.955i)T \)
67 \( 1 + (-0.920 + 0.390i)T \)
71 \( 1 + (0.480 - 0.876i)T \)
73 \( 1 + (-0.741 - 0.670i)T \)
79 \( 1 + (0.944 + 0.328i)T \)
83 \( 1 + (-0.420 - 0.907i)T \)
89 \( 1 + (-0.944 + 0.328i)T \)
97 \( 1 + (-0.645 + 0.763i)T \)
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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

−25.690419290035094251671165834228, −25.103304536976984665022997132271, −23.77907841555839420656322898524, −22.95599237154495825906541683752, −22.788865541549919496369324677222, −21.17123726986053953333449872028, −19.61618620543888213989586683709, −19.26205990166008420908909233078, −18.25416366575648239689828330415, −17.640860429606924053435104533896, −16.59159407510327590315845204283, −15.92301609106295175809507082191, −14.75406427238313343450520359312, −13.79196761830738510374853314445, −12.48278954808414740614087185061, −11.44841138228821784454662635726, −10.459276286401106692093760678512, −9.91101527019559898655367672294, −8.314945445039336897371399914540, −7.29093820343346672913380984861, −6.52110194853302025472814368384, −6.02328835175548870146438329792, −4.233810779530517393892500233767, −2.50376115663121251004250173153, −1.0935322752645216990582322728, 0.25482409974443452409172238527, 1.27685565784306585284539452987, 3.27455077551441551553308859073, 4.070461948191391907439924120196, 5.68157516084385898612233320044, 6.48048575676896495482308637664, 8.174272003904186665514065151451, 8.98122432561486745188575787703, 9.7886339314900688912124677115, 10.78416542094358684558057897925, 11.8396335690548205599086327859, 12.37639588630981925503245464386, 13.461076598711831183871693726660, 15.412382641053829745252119292119, 16.15966826474753933435693799807, 16.646851596351499817436226669168, 17.516495690184048481392532010950, 18.71464874503363016951114663459, 19.4155964570103766345345692082, 20.54186469567792088709501910858, 21.27627013464634319938562342682, 21.99511507157346445539925213783, 23.101306360333785633744376422112, 24.07393092941026410786564170319, 25.30936198738626157271777886781

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