L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.864 + 0.628i)3-s + (−0.809 + 0.587i)4-s + (−0.294 + 0.906i)5-s + (0.330 − 1.01i)6-s + (−3.77 + 2.73i)7-s + (0.809 + 0.587i)8-s + (−0.574 − 1.76i)9-s + 0.953·10-s + (−2.36 + 2.32i)11-s − 1.06·12-s + (1.17 + 3.60i)13-s + (3.77 + 2.73i)14-s + (−0.824 + 0.598i)15-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.499 + 0.362i)3-s + (−0.404 + 0.293i)4-s + (−0.131 + 0.405i)5-s + (0.134 − 0.414i)6-s + (−1.42 + 1.03i)7-s + (0.286 + 0.207i)8-s + (−0.191 − 0.589i)9-s + 0.301·10-s + (−0.714 + 0.700i)11-s − 0.308·12-s + (0.324 + 0.999i)13-s + (1.00 + 0.732i)14-s + (−0.212 + 0.154i)15-s + (0.0772 − 0.237i)16-s + (−0.0749 + 0.230i)17-s + ⋯ |
Λ(s)=(=(374s/2ΓC(s)L(s)(−0.101−0.994i)Λ(2−s)
Λ(s)=(=(374s/2ΓC(s+1/2)L(s)(−0.101−0.994i)Λ(1−s)
Degree: |
2 |
Conductor: |
374
= 2⋅11⋅17
|
Sign: |
−0.101−0.994i
|
Analytic conductor: |
2.98640 |
Root analytic conductor: |
1.72812 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ374(137,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 374, ( :1/2), −0.101−0.994i)
|
Particular Values
L(1) |
≈ |
0.504889+0.558923i |
L(21) |
≈ |
0.504889+0.558923i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.309+0.951i)T |
| 11 | 1+(2.36−2.32i)T |
| 17 | 1+(0.309−0.951i)T |
good | 3 | 1+(−0.864−0.628i)T+(0.927+2.85i)T2 |
| 5 | 1+(0.294−0.906i)T+(−4.04−2.93i)T2 |
| 7 | 1+(3.77−2.73i)T+(2.16−6.65i)T2 |
| 13 | 1+(−1.17−3.60i)T+(−10.5+7.64i)T2 |
| 19 | 1+(0.744+0.541i)T+(5.87+18.0i)T2 |
| 23 | 1+7.66T+23T2 |
| 29 | 1+(−2.25+1.64i)T+(8.96−27.5i)T2 |
| 31 | 1+(−1.43−4.41i)T+(−25.0+18.2i)T2 |
| 37 | 1+(0.366−0.266i)T+(11.4−35.1i)T2 |
| 41 | 1+(−7.75−5.63i)T+(12.6+38.9i)T2 |
| 43 | 1−0.302T+43T2 |
| 47 | 1+(−3.59−2.61i)T+(14.5+44.6i)T2 |
| 53 | 1+(1.73+5.33i)T+(−42.8+31.1i)T2 |
| 59 | 1+(−6.58+4.78i)T+(18.2−56.1i)T2 |
| 61 | 1+(4.29−13.2i)T+(−49.3−35.8i)T2 |
| 67 | 1+4.07T+67T2 |
| 71 | 1+(−3.30+10.1i)T+(−57.4−41.7i)T2 |
| 73 | 1+(6.04−4.39i)T+(22.5−69.4i)T2 |
| 79 | 1+(−1.07−3.31i)T+(−63.9+46.4i)T2 |
| 83 | 1+(1.90−5.86i)T+(−67.1−48.7i)T2 |
| 89 | 1−15.3T+89T2 |
| 97 | 1+(4.33+13.3i)T+(−78.4+57.0i)T2 |
show more | |
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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
−11.75888191731138700099035589499, −10.50249385350014156007984918794, −9.695561580186681595327360118778, −9.160843619702812276776320127565, −8.266542690017854345130593971313, −6.82995941721085104159882383834, −5.94237179916219438637181545553, −4.29350565975176607710947696145, −3.21973389692345123906414072184, −2.36959359081312373501063338021,
0.49162577646795967062615664750, 2.82670091842327053794939265756, 4.05247851607060251946148273140, 5.51864019411063074955178171626, 6.43640861656677776068758388574, 7.59690554097899994362142687840, 8.106774102841191792986977005054, 9.100016646853592382864130535602, 10.25538429661832826800424915331, 10.70337944538450421768408851348