L(s) = 1 | + (−0.750 − 1.19i)2-s + (0.448 − 1.67i)3-s + (−0.874 + 1.79i)4-s − 3.56i·5-s + (−2.34 + 0.717i)6-s + (2.81 − 0.301i)8-s + (−2.59 − 1.50i)9-s + (−4.27 + 2.67i)10-s + 0.335·11-s + (2.61 + 2.26i)12-s − 3.34·13-s + (−5.96 − 1.59i)15-s + (−2.47 − 3.14i)16-s − 0.335i·17-s + (0.150 + 4.23i)18-s − 1.84i·19-s + ⋯ |
L(s) = 1 | + (−0.530 − 0.847i)2-s + (0.258 − 0.965i)3-s + (−0.437 + 0.899i)4-s − 1.59i·5-s + (−0.956 + 0.292i)6-s + (0.994 − 0.106i)8-s + (−0.865 − 0.500i)9-s + (−1.35 + 0.845i)10-s + 0.101·11-s + (0.755 + 0.655i)12-s − 0.928·13-s + (−1.53 − 0.412i)15-s + (−0.617 − 0.786i)16-s − 0.0813i·17-s + (0.0354 + 0.999i)18-s − 0.424i·19-s + ⋯ |
Λ(s)=(=(588s/2ΓC(s)L(s)(−0.755−0.655i)Λ(2−s)
Λ(s)=(=(588s/2ΓC(s+1/2)L(s)(−0.755−0.655i)Λ(1−s)
Degree: |
2 |
Conductor: |
588
= 22⋅3⋅72
|
Sign: |
−0.755−0.655i
|
Analytic conductor: |
4.69520 |
Root analytic conductor: |
2.16684 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ588(491,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 588, ( :1/2), −0.755−0.655i)
|
Particular Values
L(1) |
≈ |
0.299249+0.802027i |
L(21) |
≈ |
0.299249+0.802027i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.750+1.19i)T |
| 3 | 1+(−0.448+1.67i)T |
| 7 | 1 |
good | 5 | 1+3.56iT−5T2 |
| 11 | 1−0.335T+11T2 |
| 13 | 1+3.34T+13T2 |
| 17 | 1+0.335iT−17T2 |
| 19 | 1+1.84iT−19T2 |
| 23 | 1−4.45T+23T2 |
| 29 | 1+5.91iT−29T2 |
| 31 | 1−5.19iT−31T2 |
| 37 | 1−3.19T+37T2 |
| 41 | 1+1.45iT−41T2 |
| 43 | 1−7.49iT−43T2 |
| 47 | 1+8.91T+47T2 |
| 53 | 1+4.79iT−53T2 |
| 59 | 1−14.0T+59T2 |
| 61 | 1−0.353T+61T2 |
| 67 | 1+3.19iT−67T2 |
| 71 | 1+10.3T+71T2 |
| 73 | 1−4.69T+73T2 |
| 79 | 1+4iT−79T2 |
| 83 | 1+6.89T+83T2 |
| 89 | 1−3.87iT−89T2 |
| 97 | 1−2T+97T2 |
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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
−9.900287933909549962634619808644, −9.206596361638590755009870605791, −8.501598097836834293967461274165, −7.83325277977994722067133664905, −6.84601469135721764036948652590, −5.30947307153381941027763597087, −4.41339031241194199826627372673, −2.92199696086430359357669246936, −1.69200323861738231278646071240, −0.55805113252775934264538799508,
2.44809678025209432961904113088, 3.64686009553084169379046143680, 4.88639988946355243790710560186, 5.89128940257090269991457176447, 6.90275085219696646556011821385, 7.58731289075767848906451831079, 8.629203910919776258815569269395, 9.606368907711503880337289558729, 10.18683752021523188722057406802, 10.84629116150756918680547935406