Dirichlet series
L(s) = 1 | + (0.347 − 0.937i)2-s + (0.347 + 0.937i)3-s + (−0.758 − 0.651i)4-s + (−0.954 − 0.299i)5-s + 6-s + (−0.440 − 0.897i)7-s + (−0.874 + 0.485i)8-s + (−0.758 + 0.651i)9-s + (−0.612 + 0.790i)10-s + (−0.820 + 0.571i)11-s + (0.347 − 0.937i)12-s + (−0.758 + 0.651i)13-s + (−0.994 + 0.101i)14-s + (−0.0506 − 0.998i)15-s + (0.151 + 0.988i)16-s + (0.612 − 0.790i)17-s + ⋯ |
L(s) = 1 | + (0.347 − 0.937i)2-s + (0.347 + 0.937i)3-s + (−0.758 − 0.651i)4-s + (−0.954 − 0.299i)5-s + 6-s + (−0.440 − 0.897i)7-s + (−0.874 + 0.485i)8-s + (−0.758 + 0.651i)9-s + (−0.612 + 0.790i)10-s + (−0.820 + 0.571i)11-s + (0.347 − 0.937i)12-s + (−0.758 + 0.651i)13-s + (−0.994 + 0.101i)14-s + (−0.0506 − 0.998i)15-s + (0.151 + 0.988i)16-s + (0.612 − 0.790i)17-s + ⋯ |
Functional equation
Invariants
Degree: | \(1\) |
Conductor: | \(311\) |
Sign: | $0.971 - 0.236i$ |
Analytic conductor: | \(33.4215\) |
Root analytic conductor: | \(33.4215\) |
Motivic weight: | \(0\) |
Rational: | no |
Arithmetic: | yes |
Character: | $\chi_{311} (279, \cdot )$ |
Primitive: | yes |
Self-dual: | no |
Analytic rank: | \(0\) |
Selberg data: | \((1,\ 311,\ (1:\ ),\ 0.971 - 0.236i)\) |
Particular Values
\(L(\frac{1}{2})\) | \(\approx\) | \(1.328822652 - 0.1595775922i\) |
\(L(\frac12)\) | \(\approx\) | \(1.328822652 - 0.1595775922i\) |
\(L(1)\) | \(\approx\) | \(0.9304564387 - 0.2501864737i\) |
\(L(1)\) | \(\approx\) | \(0.9304564387 - 0.2501864737i\) |
Euler product
$p$ | $F_p(T)$ | |
---|---|---|
bad | 311 | \( 1 \) |
good | 2 | \( 1 + (0.347 - 0.937i)T \) |
3 | \( 1 + (0.347 + 0.937i)T \) | |
5 | \( 1 + (-0.954 - 0.299i)T \) | |
7 | \( 1 + (-0.440 - 0.897i)T \) | |
11 | \( 1 + (-0.820 + 0.571i)T \) | |
13 | \( 1 + (-0.758 + 0.651i)T \) | |
17 | \( 1 + (0.612 - 0.790i)T \) | |
19 | \( 1 + (0.954 - 0.299i)T \) | |
23 | \( 1 + (0.874 - 0.485i)T \) | |
29 | \( 1 + (0.994 + 0.101i)T \) | |
31 | \( 1 + (0.612 + 0.790i)T \) | |
37 | \( 1 + (0.440 - 0.897i)T \) | |
41 | \( 1 + (-0.918 + 0.394i)T \) | |
43 | \( 1 + (0.440 + 0.897i)T \) | |
47 | \( 1 + (-0.994 - 0.101i)T \) | |
53 | \( 1 + (-0.440 - 0.897i)T \) | |
59 | \( 1 + (0.440 + 0.897i)T \) | |
61 | \( 1 + (0.954 - 0.299i)T \) | |
67 | \( 1 + (0.918 + 0.394i)T \) | |
71 | \( 1 + (-0.979 + 0.201i)T \) | |
73 | \( 1 + (0.528 - 0.848i)T \) | |
79 | \( 1 + (0.918 - 0.394i)T \) | |
83 | \( 1 + (-0.0506 + 0.998i)T \) | |
89 | \( 1 + (-0.440 + 0.897i)T \) | |
97 | \( 1 + (-0.979 + 0.201i)T \) | |
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Imaginary part of the first few zeros on the critical line
−24.96794544091862613686218904331, −24.170379144211516141210523195717, −23.463590607854487607557641786805, −22.68263686028236011847219796648, −21.80666411183188146267088236175, −20.54873481032781196715539072463, −19.18823397947488206485741859149, −18.836610770204343444381333784004, −17.89801864248956572099405862445, −16.82065586791992466000575540157, −15.61395894360997479586493324654, −15.16341646071326725754462927569, −14.16691153700122285457843459885, −13.086576236980750822602577493218, −12.38406505915313077530117594475, −11.617305743725656849922712467878, −9.85658303191781345160416184711, −8.47185061853202277019846811240, −7.98966204232718479322309158432, −7.10628011164649377906791757777, −6.03558208012880741646152413991, −5.13763380812926581795032706400, −3.3779639589378150276683694447, −2.82071807094832301273231881794, −0.52782498080365812244817285555, 0.75238193880979355690292850287, 2.71110718830337096671519289844, 3.472034184861640279694384420276, 4.6679605908668995691330144727, 4.98208612261417304635930180794, 7.069123840496939415689857089069, 8.21364139193450686188203547806, 9.451376988974951943070086620274, 10.05712139114714378275615746282, 11.03793418928896755931021345481, 11.91793443698367554731775597513, 12.94316308828735930630218922871, 13.999439172983191786983902963933, 14.7811630722604782517007226931, 15.86808056104803619545110128715, 16.56104561365296152647704233316, 17.88594771006873611128199246844, 19.261007703828698375276958265600, 19.72038841663835414926155955957, 20.59186936130466266244939180261, 21.10517774670626270592670849740, 22.33912040396256575037856008254, 23.040699820031964929673386687, 23.62937620888386227088631649935, 24.93708670281040341132986484501