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Properties

Label 1-283-283.76-r1-0-0
Degree $1$
Conductor $283$
Sign $-0.273 - 0.961i$
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  + (0.997 + 0.0667i)2-s + (0.645 − 0.763i)3-s + (0.991 + 0.133i)4-s + (0.166 − 0.986i)5-s + (0.695 − 0.718i)6-s + (−0.944 − 0.328i)7-s + (0.979 + 0.199i)8-s + (−0.166 − 0.986i)9-s + (0.231 − 0.972i)10-s + (0.991 − 0.133i)11-s + (0.741 − 0.670i)12-s + (0.695 − 0.718i)13-s + (−0.920 − 0.390i)14-s + (−0.645 − 0.763i)15-s + (0.964 + 0.264i)16-s + (−0.920 + 0.390i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0667i)2-s + (0.645 − 0.763i)3-s + (0.991 + 0.133i)4-s + (0.166 − 0.986i)5-s + (0.695 − 0.718i)6-s + (−0.944 − 0.328i)7-s + (0.979 + 0.199i)8-s + (−0.166 − 0.986i)9-s + (0.231 − 0.972i)10-s + (0.991 − 0.133i)11-s + (0.741 − 0.670i)12-s + (0.695 − 0.718i)13-s + (−0.920 − 0.390i)14-s + (−0.645 − 0.763i)15-s + (0.964 + 0.264i)16-s + (−0.920 + 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.273 - 0.961i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.273 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.746726592 - 3.636501555i\)
\(L(\frac12)\) \(\approx\) \(2.746726592 - 3.636501555i\)
\(L(1)\) \(\approx\) \(2.116391417 - 1.148907842i\)
\(L(1)\) \(\approx\) \(2.116391417 - 1.148907842i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.997 + 0.0667i)T \)
3 \( 1 + (0.645 - 0.763i)T \)
5 \( 1 + (0.166 - 0.986i)T \)
7 \( 1 + (-0.944 - 0.328i)T \)
11 \( 1 + (0.991 - 0.133i)T \)
13 \( 1 + (0.695 - 0.718i)T \)
17 \( 1 + (-0.920 + 0.390i)T \)
19 \( 1 + (-0.695 + 0.718i)T \)
23 \( 1 + (0.480 + 0.876i)T \)
29 \( 1 + (0.100 - 0.994i)T \)
31 \( 1 + (0.296 - 0.955i)T \)
37 \( 1 + (0.538 - 0.842i)T \)
41 \( 1 + (-0.979 + 0.199i)T \)
43 \( 1 + (0.420 + 0.907i)T \)
47 \( 1 + (-0.695 - 0.718i)T \)
53 \( 1 + (-0.964 + 0.264i)T \)
59 \( 1 + (0.784 + 0.619i)T \)
61 \( 1 + (0.231 + 0.972i)T \)
67 \( 1 + (0.741 + 0.670i)T \)
71 \( 1 + (0.991 - 0.133i)T \)
73 \( 1 + (-0.296 - 0.955i)T \)
79 \( 1 + (0.420 - 0.907i)T \)
83 \( 1 + (0.860 + 0.509i)T \)
89 \( 1 + (-0.420 - 0.907i)T \)
97 \( 1 + (0.480 + 0.876i)T \)
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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

−25.578828372785370851781073479120, −24.95340400982853333625724058026, −23.59664660057970563788734635167, −22.48246170290131444014183896406, −22.135624345546416116932299668802, −21.39466114662617104635268043945, −20.26185400501430518927965749305, −19.4791927505194670186267661131, −18.72804558936404535201920530856, −17.032624099044217855802203527735, −15.97885313016948662254306996486, −15.34977319420824704974882202022, −14.43069425831744023303486217222, −13.799400895574895195529610722746, −12.79466685259202955128225997488, −11.419787143459063788981999462639, −10.723123831221834693923841648416, −9.63298409958821565792542844173, −8.64781255225108711200814192744, −6.727527117391847533569047671980, −6.57040808882023843219263643958, −4.89847677765838953530787045871, −3.82091525766948534624746557260, −3.03019342121341601375476989780, −2.061520224924552020865283729125, 0.899137434275271486519423309171, 2.06232354179787678673936197339, 3.47362639465495680721232233053, 4.20232877961212173628536452015, 5.92627614069095414340075138662, 6.43851184270611603046234588830, 7.72835819927136111246874087743, 8.719182796890141776196915200128, 9.83729250354489043917963479441, 11.38307662094098238545362147872, 12.3965042731295491638692315528, 13.2385947864354026722487962903, 13.46834735357639212855957719601, 14.77772255949151696529209627104, 15.67012543378326076913101841976, 16.729186671098120582939707577829, 17.531694935730708598283614390038, 19.21015340345072764056661368459, 19.76199523293642494978357685365, 20.50926159301427327939673551297, 21.367008538528365263071095275488, 22.60971677798100432754223166730, 23.34324846579013436731339580245, 24.21312776018592190024077348731, 25.143550382868867729841434320482

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