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Properties

Label 1-283-283.132-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.0849 + 0.996i$
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  + (−0.920 − 0.390i)2-s + (−0.480 − 0.876i)3-s + (0.695 + 0.718i)4-s + (0.538 + 0.842i)5-s + (0.100 + 0.994i)6-s + (−0.420 + 0.907i)7-s + (−0.359 − 0.933i)8-s + (−0.538 + 0.842i)9-s + (−0.166 − 0.986i)10-s + (0.695 − 0.718i)11-s + (0.296 − 0.955i)12-s + (0.100 + 0.994i)13-s + (0.741 − 0.670i)14-s + (0.480 − 0.876i)15-s + (−0.0334 + 0.999i)16-s + (0.741 + 0.670i)17-s + ⋯
L(s)  = 1  + (−0.920 − 0.390i)2-s + (−0.480 − 0.876i)3-s + (0.695 + 0.718i)4-s + (0.538 + 0.842i)5-s + (0.100 + 0.994i)6-s + (−0.420 + 0.907i)7-s + (−0.359 − 0.933i)8-s + (−0.538 + 0.842i)9-s + (−0.166 − 0.986i)10-s + (0.695 − 0.718i)11-s + (0.296 − 0.955i)12-s + (0.100 + 0.994i)13-s + (0.741 − 0.670i)14-s + (0.480 − 0.876i)15-s + (−0.0334 + 0.999i)16-s + (0.741 + 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0849 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0849 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.0849 + 0.996i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.0849 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4702109339 + 0.5119986059i\)
\(L(\frac12)\) \(\approx\) \(0.4702109339 + 0.5119986059i\)
\(L(1)\) \(\approx\) \(0.6261294835 + 0.002938771988i\)
\(L(1)\) \(\approx\) \(0.6261294835 + 0.002938771988i\)

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 \( 1 + (-0.480 - 0.876i)T \)
5 \( 1 + (0.538 + 0.842i)T \)
7 \( 1 + (-0.420 + 0.907i)T \)
11 \( 1 + (0.695 - 0.718i)T \)
13 \( 1 + (0.100 + 0.994i)T \)
17 \( 1 + (0.741 + 0.670i)T \)
19 \( 1 + (-0.100 - 0.994i)T \)
23 \( 1 + (0.991 + 0.133i)T \)
29 \( 1 + (-0.824 - 0.565i)T \)
31 \( 1 + (-0.231 + 0.972i)T \)
37 \( 1 + (-0.964 - 0.264i)T \)
41 \( 1 + (0.359 - 0.933i)T \)
43 \( 1 + (-0.860 - 0.509i)T \)
47 \( 1 + (-0.100 + 0.994i)T \)
53 \( 1 + (0.0334 + 0.999i)T \)
59 \( 1 + (-0.645 - 0.763i)T \)
61 \( 1 + (-0.166 + 0.986i)T \)
67 \( 1 + (0.296 + 0.955i)T \)
71 \( 1 + (0.695 - 0.718i)T \)
73 \( 1 + (0.231 + 0.972i)T \)
79 \( 1 + (-0.860 + 0.509i)T \)
83 \( 1 + (-0.997 - 0.0667i)T \)
89 \( 1 + (0.860 + 0.509i)T \)
97 \( 1 + (0.991 + 0.133i)T \)
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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

−25.336087836056093936951432480526, −24.46358573029808134628165264649, −23.15111267792187398593940330356, −22.73640454382725711891759919890, −21.16837878921405268328321244182, −20.38169792206089247570313451492, −19.99040134597741423155428424046, −18.46168634390496384243930770028, −17.41333488762004127610923533653, −16.81162571404521616367214327050, −16.358991247387404329949996423164, −15.18268596452304522767533169463, −14.31898237890713283422560346070, −12.88153768274357797412933152764, −11.75349448865557187022002406549, −10.54557083654434176220988068529, −9.83877392968191011231863102914, −9.26163192595919619446255334858, −8.00806589887085445117165446829, −6.77745522301171449008305775041, −5.70672685172570730308568857915, −4.82377253796917352965865090769, −3.39180775117456745492920397191, −1.41307243422710893280977287501, −0.331599502008208790067028634480, 1.34229851871044130512931011497, 2.344968023387191290194946206062, 3.38367779670798575742232803519, 5.66878611718781205672235429565, 6.54090485518637955666406326712, 7.1946712081658342256226328706, 8.63883327738769301140118440546, 9.35530982008277249699587204664, 10.7059155586857100342564097231, 11.412722530008602210891317058132, 12.25063403934266997092258625539, 13.27989371013787136359135011940, 14.392217775325217648057574700576, 15.69158953656631905732172152056, 16.883156741304630414965403437603, 17.42072644786618003374486925634, 18.58517689481598624951987976191, 18.96032041920462190721177590080, 19.55956003582763904066117258291, 21.301344768163254113281983578646, 21.80802787023684360373047896543, 22.69614250127590699781778489103, 24.03702721114466694030592483899, 24.912670649796452258195296134843, 25.64446887207476331527094057169

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