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Properties

Label 1-283-283.39-r1-0-0
Degree $1$
Conductor $283$
Sign $0.613 + 0.789i$
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  + (0.166 − 0.986i)2-s + (0.824 − 0.565i)3-s + (−0.944 − 0.328i)4-s + (−0.359 − 0.933i)5-s + (−0.420 − 0.907i)6-s + (−0.741 + 0.670i)7-s + (−0.480 + 0.876i)8-s + (0.359 − 0.933i)9-s + (−0.979 + 0.199i)10-s + (−0.944 + 0.328i)11-s + (−0.964 + 0.264i)12-s + (−0.420 − 0.907i)13-s + (0.538 + 0.842i)14-s + (−0.824 − 0.565i)15-s + (0.784 + 0.619i)16-s + (0.538 − 0.842i)17-s + ⋯
L(s)  = 1  + (0.166 − 0.986i)2-s + (0.824 − 0.565i)3-s + (−0.944 − 0.328i)4-s + (−0.359 − 0.933i)5-s + (−0.420 − 0.907i)6-s + (−0.741 + 0.670i)7-s + (−0.480 + 0.876i)8-s + (0.359 − 0.933i)9-s + (−0.979 + 0.199i)10-s + (−0.944 + 0.328i)11-s + (−0.964 + 0.264i)12-s + (−0.420 − 0.907i)13-s + (0.538 + 0.842i)14-s + (−0.824 − 0.565i)15-s + (0.784 + 0.619i)16-s + (0.538 − 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.613 + 0.789i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.613 + 0.789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05271993036 + 0.02579348089i\)
\(L(\frac12)\) \(\approx\) \(-0.05271993036 + 0.02579348089i\)
\(L(1)\) \(\approx\) \(0.6232482382 - 0.6500753018i\)
\(L(1)\) \(\approx\) \(0.6232482382 - 0.6500753018i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.166 - 0.986i)T \)
3 \( 1 + (0.824 - 0.565i)T \)
5 \( 1 + (-0.359 - 0.933i)T \)
7 \( 1 + (-0.741 + 0.670i)T \)
11 \( 1 + (-0.944 + 0.328i)T \)
13 \( 1 + (-0.420 - 0.907i)T \)
17 \( 1 + (0.538 - 0.842i)T \)
19 \( 1 + (0.420 + 0.907i)T \)
23 \( 1 + (-0.892 + 0.451i)T \)
29 \( 1 + (0.860 - 0.509i)T \)
31 \( 1 + (0.0334 + 0.999i)T \)
37 \( 1 + (-0.593 + 0.805i)T \)
41 \( 1 + (0.480 + 0.876i)T \)
43 \( 1 + (0.296 + 0.955i)T \)
47 \( 1 + (0.420 - 0.907i)T \)
53 \( 1 + (-0.784 + 0.619i)T \)
59 \( 1 + (0.100 - 0.994i)T \)
61 \( 1 + (-0.979 - 0.199i)T \)
67 \( 1 + (-0.964 - 0.264i)T \)
71 \( 1 + (-0.944 + 0.328i)T \)
73 \( 1 + (-0.0334 + 0.999i)T \)
79 \( 1 + (0.296 - 0.955i)T \)
83 \( 1 + (0.231 + 0.972i)T \)
89 \( 1 + (-0.296 - 0.955i)T \)
97 \( 1 + (-0.892 + 0.451i)T \)
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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

−26.08921370736427031310369908098, −25.7116111255461870473924093210, −24.17637225412511675478985852037, −23.624398266137234708453922573825, −22.438738781657639015974738677001, −21.90335706025818115311090771025, −20.86458124883370411470992752368, −19.424991516901655943948074248, −19.03129314803827787494467592084, −17.83390040139164585587087872712, −16.544805684652459230683859695413, −15.90626645792020536289916713912, −15.13093790280870283016285677062, −14.113545375413696389832469912516, −13.680615105596415876513138402738, −12.441911509771856312045060561766, −10.70401588503124530189715191807, −9.98412144630121755234802875593, −8.91898956894396250609629514666, −7.74335709899766162728547735960, −7.15599018160234318526890738602, −5.965990552877851730273520665727, −4.47990076680366071621607672063, −3.65498468430040654897465636712, −2.66067831686866138539203339113, 0.01554385172694075623479603226, 1.31703135265236207405888723088, 2.65369470335649772781664525367, 3.38392756558544322001177416581, 4.819327979958776863875363204, 5.84009222505645285193816586302, 7.709631580892965006996428504369, 8.37682993458065949014477217474, 9.52344207804297965370871374398, 10.087021927737749952040528677391, 11.969797872648178988135985028160, 12.363882281601791232996420159738, 13.1324587676772770341891189282, 14.00873249928403154933109397511, 15.2481531915396462582100347275, 16.068856695890164658162006359298, 17.70072554956478375836122637288, 18.457312054680866333729522554191, 19.33009611514701903864134180828, 20.087030931251862289539393335900, 20.69260907291533068802606514128, 21.57386480450221473553796828320, 22.893161513267411057173192604958, 23.48708818667201118723556884167, 24.68763783674313147272419466193

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