L(s) = 1 | + (0.190 + 0.330i)2-s + (−1.11 + 1.93i)3-s + (0.927 − 1.60i)4-s + (1.11 + 1.93i)5-s − 0.854·6-s + (−2 + 1.73i)7-s + 1.47·8-s + (−1 − 1.73i)9-s + (−0.427 + 0.739i)10-s + (1.5 − 2.59i)11-s + (2.07 + 3.59i)12-s − 13-s + (−0.954 − 0.330i)14-s − 5.00·15-s + (−1.57 − 2.72i)16-s + (3.73 − 6.47i)17-s + ⋯ |
L(s) = 1 | + (0.135 + 0.233i)2-s + (−0.645 + 1.11i)3-s + (0.463 − 0.802i)4-s + (0.499 + 0.866i)5-s − 0.348·6-s + (−0.755 + 0.654i)7-s + 0.520·8-s + (−0.333 − 0.577i)9-s + (−0.135 + 0.233i)10-s + (0.452 − 0.783i)11-s + (0.598 + 1.03i)12-s − 0.277·13-s + (−0.255 − 0.0884i)14-s − 1.29·15-s + (−0.393 − 0.681i)16-s + (0.906 − 1.56i)17-s + ⋯ |
Λ(s)=(=(91s/2ΓC(s)L(s)(0.386−0.922i)Λ(2−s)
Λ(s)=(=(91s/2ΓC(s+1/2)L(s)(0.386−0.922i)Λ(1−s)
Degree: |
2 |
Conductor: |
91
= 7⋅13
|
Sign: |
0.386−0.922i
|
Analytic conductor: |
0.726638 |
Root analytic conductor: |
0.852431 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ91(53,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 91, ( :1/2), 0.386−0.922i)
|
Particular Values
L(1) |
≈ |
0.821157+0.546219i |
L(21) |
≈ |
0.821157+0.546219i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1+(2−1.73i)T |
| 13 | 1+T |
good | 2 | 1+(−0.190−0.330i)T+(−1+1.73i)T2 |
| 3 | 1+(1.11−1.93i)T+(−1.5−2.59i)T2 |
| 5 | 1+(−1.11−1.93i)T+(−2.5+4.33i)T2 |
| 11 | 1+(−1.5+2.59i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−3.73+6.47i)T+(−8.5−14.7i)T2 |
| 19 | 1+(1.5+2.59i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−1.88−3.25i)T+(−11.5+19.9i)T2 |
| 29 | 1+4.47T+29T2 |
| 31 | 1+(2.5−4.33i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−4.35−7.54i)T+(−18.5+32.0i)T2 |
| 41 | 1−4.47T+41T2 |
| 43 | 1+8T+43T2 |
| 47 | 1+(0.736+1.27i)T+(−23.5+40.7i)T2 |
| 53 | 1+(0.736−1.27i)T+(−26.5−45.8i)T2 |
| 59 | 1+(3.73−6.47i)T+(−29.5−51.0i)T2 |
| 61 | 1+(1.5+2.59i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−1.5+2.59i)T+(−33.5−58.0i)T2 |
| 71 | 1−8.94T+71T2 |
| 73 | 1+(5.35−9.27i)T+(−36.5−63.2i)T2 |
| 79 | 1+(5.35+9.27i)T+(−39.5+68.4i)T2 |
| 83 | 1+83T2 |
| 89 | 1+(−1.11−1.93i)T+(−44.5+77.0i)T2 |
| 97 | 1+17.4T+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
−14.56454761009922008723439693998, −13.54874918285554397979028465060, −11.71978116591483988392727164869, −10.98317839457928490395678231713, −9.963985434852701662601982436292, −9.365255247777837370764012818202, −7.00073560932273874586189697192, −5.96738277904419845478407978227, −5.09779491066717041247569614310, −3.00255312100191105218792907301,
1.69057622283079173193073362078, 3.97646047461772695807843743915, 5.94049446538056836979371913842, 6.98263698841547473536537244743, 7.974674709273874811639719904103, 9.576703550189450260791544118253, 10.91420614036800776582806184832, 12.30857092311281099427437521250, 12.68174269319074604724993730510, 13.21773273479029602439484288605