↑

Properties

Label 1-283-283.148-r1-0-0
Degree $1$
Conductor $283$
Sign $0.496 + 0.868i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.920 − 0.390i)2-s + (−0.519 + 0.854i)3-s + (0.695 + 0.718i)4-s + (−0.999 + 0.0445i)5-s + (0.811 − 0.584i)6-s + (−0.575 − 0.818i)7-s + (−0.359 − 0.933i)8-s + (−0.460 − 0.887i)9-s + (0.937 + 0.348i)10-s + (0.274 + 0.961i)11-s + (−0.975 + 0.220i)12-s + (−0.911 − 0.410i)13-s + (0.210 + 0.977i)14-s + (0.480 − 0.876i)15-s + (−0.0334 + 0.999i)16-s + (−0.951 + 0.306i)17-s + ⋯
L(s)  = 1  + (−0.920 − 0.390i)2-s + (−0.519 + 0.854i)3-s + (0.695 + 0.718i)4-s + (−0.999 + 0.0445i)5-s + (0.811 − 0.584i)6-s + (−0.575 − 0.818i)7-s + (−0.359 − 0.933i)8-s + (−0.460 − 0.887i)9-s + (0.937 + 0.348i)10-s + (0.274 + 0.961i)11-s + (−0.975 + 0.220i)12-s + (−0.911 − 0.410i)13-s + (0.210 + 0.977i)14-s + (0.480 − 0.876i)15-s + (−0.0334 + 0.999i)16-s + (−0.951 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.496 + 0.868i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.496 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2179393282 + 0.1264765580i\)
\(L(\frac12)\) \(\approx\) \(0.2179393282 + 0.1264765580i\)
\(L(1)\) \(\approx\) \(0.3795324184 + 9.897205471\times10^{-5}i\)
\(L(1)\) \(\approx\) \(0.3795324184 + 9.897205471\times10^{-5}i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.920 - 0.390i)T \)
3 \( 1 + (-0.519 + 0.854i)T \)
5 \( 1 + (-0.999 + 0.0445i)T \)
7 \( 1 + (-0.575 - 0.818i)T \)
11 \( 1 + (0.274 + 0.961i)T \)
13 \( 1 + (-0.911 - 0.410i)T \)
17 \( 1 + (-0.951 + 0.306i)T \)
19 \( 1 + (-0.100 - 0.994i)T \)
23 \( 1 + (-0.380 - 0.924i)T \)
29 \( 1 + (-0.824 - 0.565i)T \)
31 \( 1 + (0.958 - 0.285i)T \)
37 \( 1 + (0.711 - 0.703i)T \)
41 \( 1 + (-0.987 + 0.155i)T \)
43 \( 1 + (-0.860 - 0.509i)T \)
47 \( 1 + (-0.811 - 0.584i)T \)
53 \( 1 + (0.0334 + 0.999i)T \)
59 \( 1 + (0.984 - 0.177i)T \)
61 \( 1 + (-0.166 + 0.986i)T \)
67 \( 1 + (0.296 + 0.955i)T \)
71 \( 1 + (0.695 - 0.718i)T \)
73 \( 1 + (-0.958 - 0.285i)T \)
79 \( 1 + (-0.860 + 0.509i)T \)
83 \( 1 + (0.441 + 0.897i)T \)
89 \( 1 + (-0.871 + 0.490i)T \)
97 \( 1 + (-0.610 + 0.791i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.99535981260460914286154228490, −24.420005111882613865891138558594, −23.72889044099482177246504502107, −22.71176520472444508951311637415, −21.78585855702199668398684217391, −20.10980784481553164572383707492, −19.325356695587632802366475997, −18.889629604204919555112051048092, −18.04595435226047802854048128763, −16.84898995024523840036202680610, −16.26260847334308369254539752496, −15.33865379035185833529171451105, −14.23478551889175984561028362345, −12.85004023691490216015860856687, −11.63752818971894083923273538191, −11.485995610906306166893862791680, −9.9562367724535899022412394934, −8.7279568182091002865992006070, −8.00404588457785164067932792039, −6.94080976679399582694706877946, −6.187100122527968152749329254341, −5.05640753734721181063431110581, −3.085338040821363359839003160198, −1.73671421651702541355625124211, −0.230047811852137355809151780478, 0.525110230050993997665521654327, 2.6116445451617928174555689957, 3.895970815307937251602577008222, 4.5852536552593568992654002444, 6.57486740580947216534486406093, 7.27402383274180016777556390935, 8.52553215268567925865364244298, 9.65507359233819148310534432577, 10.3260449986805398386648278478, 11.23498407629791033612925232468, 12.06403353879513361477679623920, 13.00122996007262009481450823769, 14.977720601077097142735371343348, 15.48414046506760359157916091043, 16.54419329640993423700456460131, 17.16264882622686340361232106719, 18.02501424498691666237095561958, 19.42870191575569173817573229020, 20.02703721020746710976338451628, 20.57179821376085375044084289533, 22.03741871248858317301358428544, 22.540348585523496345619039257080, 23.58319804633205727923285644787, 24.68866532772688228117919696789, 26.09924609713477843499439903608

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