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Properties

Label 1-283-283.72-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.279 + 0.960i$
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  + (0.892 + 0.451i)2-s + (−0.317 − 0.948i)3-s + (0.593 + 0.805i)4-s + (0.122 − 0.992i)5-s + (0.144 − 0.989i)6-s + (0.274 + 0.961i)7-s + (0.166 + 0.986i)8-s + (−0.798 + 0.602i)9-s + (0.556 − 0.830i)10-s + (0.400 + 0.916i)11-s + (0.575 − 0.818i)12-s + (−0.929 + 0.369i)13-s + (−0.188 + 0.982i)14-s + (−0.979 + 0.199i)15-s + (−0.296 + 0.955i)16-s + (−0.756 + 0.654i)17-s + ⋯
L(s)  = 1  + (0.892 + 0.451i)2-s + (−0.317 − 0.948i)3-s + (0.593 + 0.805i)4-s + (0.122 − 0.992i)5-s + (0.144 − 0.989i)6-s + (0.274 + 0.961i)7-s + (0.166 + 0.986i)8-s + (−0.798 + 0.602i)9-s + (0.556 − 0.830i)10-s + (0.400 + 0.916i)11-s + (0.575 − 0.818i)12-s + (−0.929 + 0.369i)13-s + (−0.188 + 0.982i)14-s + (−0.979 + 0.199i)15-s + (−0.296 + 0.955i)16-s + (−0.756 + 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.279 + 0.960i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.279 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.195171376 + 1.592740826i\)
\(L(\frac12)\) \(\approx\) \(1.195171376 + 1.592740826i\)
\(L(1)\) \(\approx\) \(1.390791613 + 0.3293145140i\)
\(L(1)\) \(\approx\) \(1.390791613 + 0.3293145140i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.892 + 0.451i)T \)
3 \( 1 + (-0.317 - 0.948i)T \)
5 \( 1 + (0.122 - 0.992i)T \)
7 \( 1 + (0.274 + 0.961i)T \)
11 \( 1 + (0.400 + 0.916i)T \)
13 \( 1 + (-0.929 + 0.369i)T \)
17 \( 1 + (-0.756 + 0.654i)T \)
19 \( 1 + (-0.784 - 0.619i)T \)
23 \( 1 + (0.628 - 0.777i)T \)
29 \( 1 + (-0.645 + 0.763i)T \)
31 \( 1 + (-0.0111 + 0.999i)T \)
37 \( 1 + (0.210 + 0.977i)T \)
41 \( 1 + (0.937 - 0.348i)T \)
43 \( 1 + (-0.100 + 0.994i)T \)
47 \( 1 + (-0.144 - 0.989i)T \)
53 \( 1 + (0.296 + 0.955i)T \)
59 \( 1 + (0.882 + 0.470i)T \)
61 \( 1 + (-0.997 + 0.0667i)T \)
67 \( 1 + (0.420 - 0.907i)T \)
71 \( 1 + (0.593 - 0.805i)T \)
73 \( 1 + (0.0111 + 0.999i)T \)
79 \( 1 + (-0.100 - 0.994i)T \)
83 \( 1 + (-0.0779 + 0.996i)T \)
89 \( 1 + (0.811 + 0.584i)T \)
97 \( 1 + (-0.987 - 0.155i)T \)
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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

−24.93373843734590759413889519757, −23.884942005504851529354604275754, −22.87075528646381345608229893288, −22.4483818875057836765208626028, −21.55521497116530792230010948497, −20.84941358347277021352194137152, −19.82621659015475398918358134719, −19.03299021991120674379780382016, −17.58927703855517172807088225874, −16.71865884787996562926597881500, −15.59252607134299049406661053677, −14.70437188820925766264032013622, −14.149245287394577289942980841593, −13.112203640216960674104694550751, −11.52711178508902339850606668288, −11.11809920787840208033928651728, −10.2701519086238304157185485982, −9.42423272968090854039997707044, −7.53092172854550706124340833386, −6.42017321053941140633314836594, −5.48093915424531875885483834159, −4.25276774092281068125819545718, −3.53500613661621579910737364440, −2.3827968165528851955064620069, −0.42687516816569639455228594295, 1.703596791886350779254065886109, 2.508971971904033817770475034325, 4.52214446566679696862909866343, 5.11988284081263500185297646019, 6.27776455406743653792520860067, 7.10612946848608791485081863238, 8.32004826881294406465657193024, 9.07071399596769416855823415524, 11.03781441523720110908985383546, 12.1706914744115534795453301912, 12.49688934491616771286164846846, 13.288775183918166750883845768574, 14.5495132335467161117418817696, 15.244519140133953559134096727652, 16.576559315126734718161235556144, 17.23992958333671757890517537420, 17.99930505460290205694468275754, 19.469535158273750309880568645228, 20.15844918732342507203178127528, 21.38796941510237425780728300244, 22.08175621887305175911602618089, 23.07964526800999892470249833572, 24.054243514496217121045505222090, 24.5324657799428441365597698681, 25.1701984711313922157207094033

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