L(s) = 1 | + (−1.30 + 0.750i)2-s + (−0.0792 + 2.99i)3-s + (−0.873 + 1.51i)4-s + (−6.06 − 3.50i)5-s + (−2.14 − 3.95i)6-s + (0.489 + 0.847i)7-s − 8.62i·8-s + (−8.98 − 0.475i)9-s + 10.5·10-s + (12.1 − 7.00i)11-s + (−4.46 − 2.73i)12-s + (−1.80 + 3.12i)13-s + (−1.27 − 0.734i)14-s + (10.9 − 17.9i)15-s + (2.98 + 5.16i)16-s + 4.79i·17-s + ⋯ |
L(s) = 1 | + (−0.650 + 0.375i)2-s + (−0.0264 + 0.999i)3-s + (−0.218 + 0.378i)4-s + (−1.21 − 0.700i)5-s + (−0.357 − 0.659i)6-s + (0.0699 + 0.121i)7-s − 1.07i·8-s + (−0.998 − 0.0528i)9-s + 1.05·10-s + (1.10 − 0.637i)11-s + (−0.372 − 0.228i)12-s + (−0.138 + 0.240i)13-s + (−0.0909 − 0.0524i)14-s + (0.732 − 1.19i)15-s + (0.186 + 0.322i)16-s + 0.282i·17-s + ⋯ |
Λ(s)=(=(117s/2ΓC(s)L(s)(−0.391+0.920i)Λ(3−s)
Λ(s)=(=(117s/2ΓC(s+1)L(s)(−0.391+0.920i)Λ(1−s)
Degree: |
2 |
Conductor: |
117
= 32⋅13
|
Sign: |
−0.391+0.920i
|
Analytic conductor: |
3.18801 |
Root analytic conductor: |
1.78550 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ117(14,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 117, ( :1), −0.391+0.920i)
|
Particular Values
L(23) |
≈ |
0.00926876−0.0140111i |
L(21) |
≈ |
0.00926876−0.0140111i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.0792−2.99i)T |
| 13 | 1+(1.80−3.12i)T |
good | 2 | 1+(1.30−0.750i)T+(2−3.46i)T2 |
| 5 | 1+(6.06+3.50i)T+(12.5+21.6i)T2 |
| 7 | 1+(−0.489−0.847i)T+(−24.5+42.4i)T2 |
| 11 | 1+(−12.1+7.00i)T+(60.5−104.i)T2 |
| 17 | 1−4.79iT−289T2 |
| 19 | 1+29.6T+361T2 |
| 23 | 1+(38.6+22.3i)T+(264.5+458.i)T2 |
| 29 | 1+(40.2−23.2i)T+(420.5−728.i)T2 |
| 31 | 1+(1.40−2.43i)T+(−480.5−832.i)T2 |
| 37 | 1−9.83T+1.36e3T2 |
| 41 | 1+(11.8+6.85i)T+(840.5+1.45e3i)T2 |
| 43 | 1+(−38.8−67.2i)T+(−924.5+1.60e3i)T2 |
| 47 | 1+(19.6−11.3i)T+(1.10e3−1.91e3i)T2 |
| 53 | 1−35.0iT−2.80e3T2 |
| 59 | 1+(57.8+33.4i)T+(1.74e3+3.01e3i)T2 |
| 61 | 1+(20.1+34.8i)T+(−1.86e3+3.22e3i)T2 |
| 67 | 1+(−9.77+16.9i)T+(−2.24e3−3.88e3i)T2 |
| 71 | 1−84.0iT−5.04e3T2 |
| 73 | 1+6.25T+5.32e3T2 |
| 79 | 1+(−1.65−2.86i)T+(−3.12e3+5.40e3i)T2 |
| 83 | 1+(−73.3+42.3i)T+(3.44e3−5.96e3i)T2 |
| 89 | 1−60.2iT−7.92e3T2 |
| 97 | 1+(9.35+16.2i)T+(−4.70e3+8.14e3i)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
−14.31409713682285939523113109913, −12.70008400796109159959655549475, −11.88081260738779772339232106912, −10.81662788905081735905231952386, −9.407958258485156983807701159453, −8.631847722706881886820542279386, −7.995813698062716928610363685937, −6.31742888650884184846775252236, −4.37611914280329290387979176551, −3.79071844123557238769233111201,
0.01460482790123195742642921086, 1.99524284511032688292320699900, 4.02312055163835756605612756502, 6.01985434686661234776880117903, 7.29653622621147570933216040449, 8.120204946515953368099227601078, 9.320662560797103146487722483282, 10.66160639357385425528614936090, 11.55806562911771442427510790826, 12.20143824536747833742867533875