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Properties

Label 1-283-283.7-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  + (−0.0334 − 0.999i)2-s + (0.996 − 0.0890i)3-s + (−0.997 + 0.0667i)4-s + (−0.338 − 0.940i)5-s + (−0.122 − 0.992i)6-s + (0.937 + 0.348i)7-s + (0.100 + 0.994i)8-s + (0.984 − 0.177i)9-s + (−0.929 + 0.369i)10-s + (0.556 − 0.830i)11-s + (−0.987 + 0.155i)12-s + (−0.798 + 0.602i)13-s + (0.317 − 0.948i)14-s + (−0.420 − 0.907i)15-s + (0.991 − 0.133i)16-s + (0.662 − 0.749i)17-s + ⋯
L(s)  = 1  + (−0.0334 − 0.999i)2-s + (0.996 − 0.0890i)3-s + (−0.997 + 0.0667i)4-s + (−0.338 − 0.940i)5-s + (−0.122 − 0.992i)6-s + (0.937 + 0.348i)7-s + (0.100 + 0.994i)8-s + (0.984 − 0.177i)9-s + (−0.929 + 0.369i)10-s + (0.556 − 0.830i)11-s + (−0.987 + 0.155i)12-s + (−0.798 + 0.602i)13-s + (0.317 − 0.948i)14-s + (−0.420 − 0.907i)15-s + (0.991 − 0.133i)16-s + (0.662 − 0.749i)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: $\chi_{283} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.381 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8982671554 - 1.342724768i\)
\(L(\frac12)\) \(\approx\) \(0.8982671554 - 1.342724768i\)
\(L(1)\) \(\approx\) \(1.065490220 - 0.8379893879i\)
\(L(1)\) \(\approx\) \(1.065490220 - 0.8379893879i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.0334 - 0.999i)T \)
3 \( 1 + (0.996 - 0.0890i)T \)
5 \( 1 + (-0.338 - 0.940i)T \)
7 \( 1 + (0.937 + 0.348i)T \)
11 \( 1 + (0.556 - 0.830i)T \)
13 \( 1 + (-0.798 + 0.602i)T \)
17 \( 1 + (0.662 - 0.749i)T \)
19 \( 1 + (0.920 + 0.390i)T \)
23 \( 1 + (-0.871 - 0.490i)T \)
29 \( 1 + (-0.741 - 0.670i)T \)
31 \( 1 + (-0.993 - 0.111i)T \)
37 \( 1 + (0.519 + 0.854i)T \)
41 \( 1 + (-0.911 + 0.410i)T \)
43 \( 1 + (-0.538 - 0.842i)T \)
47 \( 1 + (-0.122 + 0.992i)T \)
53 \( 1 + (0.991 + 0.133i)T \)
59 \( 1 + (0.188 - 0.982i)T \)
61 \( 1 + (0.784 - 0.619i)T \)
67 \( 1 + (0.359 + 0.933i)T \)
71 \( 1 + (-0.997 - 0.0667i)T \)
73 \( 1 + (-0.993 + 0.111i)T \)
79 \( 1 + (-0.538 + 0.842i)T \)
83 \( 1 + (-0.711 - 0.703i)T \)
89 \( 1 + (0.999 - 0.0445i)T \)
97 \( 1 + (0.0111 + 0.999i)T \)
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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

−25.88751537042915752447365244106, −25.06312222215801991323003870771, −24.195297721903958144196181861265, −23.39726763374379952527691296380, −22.26002194888542500958022423683, −21.61780514412352545775799165917, −20.14906874426923127243706276852, −19.54777944121603243850362386984, −18.27881520231215123984112238265, −17.80753486951676216902324663878, −16.57844157281317653182810504544, −15.3350287117665409080323917509, −14.70085432629376047281967992000, −14.36488199294625025738074736318, −13.22192517944319178569016391669, −11.96582869405059152806735498550, −10.40138258123668223010834691587, −9.68087939275754709164961301673, −8.437659812911465393777906212143, −7.351236156861552634653402724484, −7.28631698726725052072003911906, −5.49544736790736473926369396138, −4.25967911226473534336433463089, −3.39930541953933860665593321585, −1.75302925146026159044185679681, 1.16380674397073670732668457084, 2.15096493528681939509047062052, 3.48275341380138602217518742269, 4.430372539265855576248254671878, 5.42180677062902671475250960976, 7.59633771570610969939357623361, 8.34647952934090677598080111549, 9.16601665038379440568050494436, 9.92162242103425053688535863951, 11.62391327131969609176442932720, 11.94945108683914236364300587588, 13.15968735449227098353101118967, 14.10292921030799078392431071659, 14.67852858710779938711535525445, 16.16308609295765774079219487112, 17.14149403358691088157870147364, 18.48911533815793229338617144721, 18.975595456429116678577149310726, 20.20040847025598984915571175865, 20.44924707955429442131852021621, 21.45770504588063016060317249766, 22.151701537198308145218848420907, 23.75768298892036193666576147170, 24.333088432873696153301650968695, 25.151324094776830064179125048534

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