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Properties

Label 1-283-283.36-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.669 - 0.742i$
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  + (−0.892 + 0.451i)2-s + (0.662 + 0.749i)3-s + (0.593 − 0.805i)4-s + (−0.798 + 0.602i)5-s + (−0.929 − 0.369i)6-s + (−0.970 + 0.242i)7-s + (−0.166 + 0.986i)8-s + (−0.122 + 0.992i)9-s + (0.441 − 0.897i)10-s + (−0.993 + 0.111i)11-s + (0.996 − 0.0890i)12-s + (0.144 + 0.989i)13-s + (0.756 − 0.654i)14-s + (−0.979 − 0.199i)15-s + (−0.296 − 0.955i)16-s + (0.188 − 0.982i)17-s + ⋯
L(s)  = 1  + (−0.892 + 0.451i)2-s + (0.662 + 0.749i)3-s + (0.593 − 0.805i)4-s + (−0.798 + 0.602i)5-s + (−0.929 − 0.369i)6-s + (−0.970 + 0.242i)7-s + (−0.166 + 0.986i)8-s + (−0.122 + 0.992i)9-s + (0.441 − 0.897i)10-s + (−0.993 + 0.111i)11-s + (0.996 − 0.0890i)12-s + (0.144 + 0.989i)13-s + (0.756 − 0.654i)14-s + (−0.979 − 0.199i)15-s + (−0.296 − 0.955i)16-s + (0.188 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.669 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1058607286 + 0.2379773924i\)
\(L(\frac12)\) \(\approx\) \(-0.1058607286 + 0.2379773924i\)
\(L(1)\) \(\approx\) \(0.4203348785 + 0.3251012109i\)
\(L(1)\) \(\approx\) \(0.4203348785 + 0.3251012109i\)

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 \( 1 + (0.662 + 0.749i)T \)
5 \( 1 + (-0.798 + 0.602i)T \)
7 \( 1 + (-0.970 + 0.242i)T \)
11 \( 1 + (-0.993 + 0.111i)T \)
13 \( 1 + (0.144 + 0.989i)T \)
17 \( 1 + (0.188 - 0.982i)T \)
19 \( 1 + (0.784 - 0.619i)T \)
23 \( 1 + (-0.987 + 0.155i)T \)
29 \( 1 + (-0.645 - 0.763i)T \)
31 \( 1 + (-0.871 - 0.490i)T \)
37 \( 1 + (0.951 - 0.306i)T \)
41 \( 1 + (-0.770 + 0.637i)T \)
43 \( 1 + (0.100 + 0.994i)T \)
47 \( 1 + (-0.929 + 0.369i)T \)
53 \( 1 + (-0.296 + 0.955i)T \)
59 \( 1 + (-0.848 - 0.528i)T \)
61 \( 1 + (-0.997 - 0.0667i)T \)
67 \( 1 + (-0.420 - 0.907i)T \)
71 \( 1 + (0.593 + 0.805i)T \)
73 \( 1 + (-0.871 + 0.490i)T \)
79 \( 1 + (0.100 - 0.994i)T \)
83 \( 1 + (0.902 + 0.431i)T \)
89 \( 1 + (-0.911 - 0.410i)T \)
97 \( 1 + (0.628 + 0.777i)T \)
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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

−25.30240891243836492901293825315, −24.20208887492427994879343157394, −23.46550366348228555417241125795, −22.24698476959805997591683827989, −20.8309570342519993594462140017, −20.138708804600252506454982239995, −19.662929960337835448652893150272, −18.68326381935179940882444884944, −18.06373079382555583429201498412, −16.75070654552869708685832236854, −15.960662108366010436572631961759, −15.08518459356713864416759691976, −13.388205586241288839265797009720, −12.69558507696450107415535266179, −12.12800094044929053503110903488, −10.70271738164136520850651433279, −9.760042920602213678523897407805, −8.610807196730626766651981784266, −7.9273276181689754436293961603, −7.21329484241600404013460329763, −5.81770676264896957953611162927, −3.69268571359218656494953241956, −3.11766981099389685720497052709, −1.60199537314958173040654319544, −0.20569105452419529878832071494, 2.363289767425075404199178234976, 3.25085467554619528520407639012, 4.686599881927053152837517263569, 6.07431785176237957294187609843, 7.339157363925722302321227477476, 7.95274087399736274972840079723, 9.30822476358367598603431798265, 9.741025928773507970049872247255, 10.901841057501184209818410944580, 11.71917938896726778602055597150, 13.45045930971852145431638721143, 14.461169592410425090366256709423, 15.44030629763330132774530172493, 15.99111255216998966318691509847, 16.55182359522725632990648135035, 18.30796612617167060924646956295, 18.75323336608197982678077096635, 19.76617071844366577687260460234, 20.3153346605190184222654209463, 21.56928353736853763322085377492, 22.62436132167579265254390058516, 23.515211641871167448360767087338, 24.580315073877050784209305223, 25.70484181443396441586907026824, 26.29554037770353991260949175457

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