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
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: | $\chi_{283} (108, \cdot )$ |
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
$p$ | $F_p(T)$ | |
---|---|---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (0.997 - 0.0667i)T \) |
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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Imaginary part of the first few zeros on the critical line
−25.143550382868867729841434320482, −24.21312776018592190024077348731, −23.34324846579013436731339580245, −22.60971677798100432754223166730, −21.367008538528365263071095275488, −20.50926159301427327939673551297, −19.76199523293642494978357685365, −19.21015340345072764056661368459, −17.531694935730708598283614390038, −16.729186671098120582939707577829, −15.67012543378326076913101841976, −14.77772255949151696529209627104, −13.46834735357639212855957719601, −13.2385947864354026722487962903, −12.3965042731295491638692315528, −11.38307662094098238545362147872, −9.83729250354489043917963479441, −8.719182796890141776196915200128, −7.72835819927136111246874087743, −6.43851184270611603046234588830, −5.92627614069095414340075138662, −4.20232877961212173628536452015, −3.47362639465495680721232233053, −2.06232354179787678673936197339, −0.899137434275271486519423309171, 2.061520224924552020865283729125, 3.03019342121341601375476989780, 3.82091525766948534624746557260, 4.89847677765838953530787045871, 6.57040808882023843219263643958, 6.727527117391847533569047671980, 8.64781255225108711200814192744, 9.63298409958821565792542844173, 10.723123831221834693923841648416, 11.419787143459063788981999462639, 12.79466685259202955128225997488, 13.799400895574895195529610722746, 14.43069425831744023303486217222, 15.34977319420824704974882202022, 15.97885313016948662254306996486, 17.032624099044217855802203527735, 18.72804558936404535201920530856, 19.4791927505194670186267661131, 20.26185400501430518927965749305, 21.39466114662617104635268043945, 22.135624345546416116932299668802, 22.48246170290131444014183896406, 23.59664660057970563788734635167, 24.95340400982853333625724058026, 25.578828372785370851781073479120