L(s) = 1 | + (−2.52 + 1.04i)3-s + (−1.32 − 3.20i)5-s + (−1.04 + 2.52i)7-s + (3.15 − 3.15i)9-s + (−1.17 − 0.485i)11-s − 5.46i·13-s + (6.69 + 6.69i)15-s + (−1.03 − 1.03i)19-s − 7.46i·21-s + (1.17 + 0.485i)23-s + (−4.94 + 4.94i)25-s + (−1.53 + 3.69i)27-s + (−1.32 − 3.20i)29-s + (3.87 − 1.60i)31-s + 3.46·33-s + ⋯ |
L(s) = 1 | + (−1.45 + 0.603i)3-s + (−0.592 − 1.43i)5-s + (−0.395 + 0.954i)7-s + (1.05 − 1.05i)9-s + (−0.353 − 0.146i)11-s − 1.51i·13-s + (1.72 + 1.72i)15-s + (−0.237 − 0.237i)19-s − 1.62i·21-s + (0.244 + 0.101i)23-s + (−0.989 + 0.989i)25-s + (−0.294 + 0.711i)27-s + (−0.246 − 0.594i)29-s + (0.696 − 0.288i)31-s + 0.603·33-s + ⋯ |
Λ(s)=(=(1156s/2ΓC(s)L(s)(−0.275−0.961i)Λ(2−s)
Λ(s)=(=(1156s/2ΓC(s+1/2)L(s)(−0.275−0.961i)Λ(1−s)
Degree: |
2 |
Conductor: |
1156
= 22⋅172
|
Sign: |
−0.275−0.961i
|
Analytic conductor: |
9.23070 |
Root analytic conductor: |
3.03820 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1156(733,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1156, ( :1/2), −0.275−0.961i)
|
Particular Values
L(1) |
≈ |
0.2672395393 |
L(21) |
≈ |
0.2672395393 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 17 | 1 |
good | 3 | 1+(2.52−1.04i)T+(2.12−2.12i)T2 |
| 5 | 1+(1.32+3.20i)T+(−3.53+3.53i)T2 |
| 7 | 1+(1.04−2.52i)T+(−4.94−4.94i)T2 |
| 11 | 1+(1.17+0.485i)T+(7.77+7.77i)T2 |
| 13 | 1+5.46iT−13T2 |
| 19 | 1+(1.03+1.03i)T+19iT2 |
| 23 | 1+(−1.17−0.485i)T+(16.2+16.2i)T2 |
| 29 | 1+(1.32+3.20i)T+(−20.5+20.5i)T2 |
| 31 | 1+(−3.87+1.60i)T+(21.9−21.9i)T2 |
| 37 | 1+(4.19−1.73i)T+(26.1−26.1i)T2 |
| 41 | 1+(2.29−5.54i)T+(−28.9−28.9i)T2 |
| 43 | 1+(5.93−5.93i)T−43iT2 |
| 47 | 1−6.92iT−47T2 |
| 53 | 1+(−9.14−9.14i)T+53iT2 |
| 59 | 1+(1.79−1.79i)T−59iT2 |
| 61 | 1+(−0.205+0.495i)T+(−43.1−43.1i)T2 |
| 67 | 1+14.9T+67T2 |
| 71 | 1+(−7.57+3.13i)T+(50.2−50.2i)T2 |
| 73 | 1+(−0.765−1.84i)T+(−51.6+51.6i)T2 |
| 79 | 1+(11.2+4.66i)T+(55.8+55.8i)T2 |
| 83 | 1+(−1.79−1.79i)T+83iT2 |
| 89 | 1−2.53iT−89T2 |
| 97 | 1+(−1.88−4.55i)T+(−68.5+68.5i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.11311443731527758316555721972, −9.285526589838840612835757235265, −8.461662361208195500944606006130, −7.72337889553738157890754393879, −6.26755024870193871321842341805, −5.63305292599271496620715196304, −5.01946548934054809249113293756, −4.32949864192438787625550620966, −2.97701346031119652947654673266, −0.935039929673078303302323424031,
0.18491438191775380294537965242, 1.89109072536710963788794387088, 3.41893587666709115152548282594, 4.31526352226140239796940678863, 5.45513487569375410357809238698, 6.56209771394876036212856362122, 6.98813373802779280243248753618, 7.26781835872995146678415975705, 8.616760478786206135674877193859, 10.08037204787091706858140123555