L(s) = 1 | + (−0.201 − 0.349i)2-s + (1.91 − 3.32i)4-s − 6.09·5-s + (−9.03 − 5.21i)7-s − 3.16·8-s + (1.22 + 2.12i)10-s + (1.06 + 1.83i)11-s + (12.9 − 0.692i)13-s + 4.21i·14-s + (−7.03 − 12.1i)16-s + (−26.4 − 15.2i)17-s + (3.26 + 1.88i)19-s + (−11.6 + 20.2i)20-s + (0.428 − 0.741i)22-s + (28.7 − 16.6i)23-s + ⋯ |
L(s) = 1 | + (−0.100 − 0.174i)2-s + (0.479 − 0.830i)4-s − 1.21·5-s + (−1.29 − 0.745i)7-s − 0.395·8-s + (0.122 + 0.212i)10-s + (0.0964 + 0.167i)11-s + (0.998 − 0.0532i)13-s + 0.300i·14-s + (−0.439 − 0.761i)16-s + (−1.55 − 0.897i)17-s + (0.171 + 0.0992i)19-s + (−0.584 + 1.01i)20-s + (0.0194 − 0.0337i)22-s + (1.25 − 0.722i)23-s + ⋯ |
Λ(s)=(=(117s/2ΓC(s)L(s)(−0.792+0.609i)Λ(3−s)
Λ(s)=(=(117s/2ΓC(s+1)L(s)(−0.792+0.609i)Λ(1−s)
Degree: |
2 |
Conductor: |
117
= 32⋅13
|
Sign: |
−0.792+0.609i
|
Analytic conductor: |
3.18801 |
Root analytic conductor: |
1.78550 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ117(62,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 117, ( :1), −0.792+0.609i)
|
Particular Values
L(23) |
≈ |
0.237725−0.698623i |
L(21) |
≈ |
0.237725−0.698623i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 13 | 1+(−12.9+0.692i)T |
good | 2 | 1+(0.201+0.349i)T+(−2+3.46i)T2 |
| 5 | 1+6.09T+25T2 |
| 7 | 1+(9.03+5.21i)T+(24.5+42.4i)T2 |
| 11 | 1+(−1.06−1.83i)T+(−60.5+104.i)T2 |
| 17 | 1+(26.4+15.2i)T+(144.5+250.i)T2 |
| 19 | 1+(−3.26−1.88i)T+(180.5+312.i)T2 |
| 23 | 1+(−28.7+16.6i)T+(264.5−458.i)T2 |
| 29 | 1+(−35.3+20.4i)T+(420.5−728.i)T2 |
| 31 | 1−21.3iT−961T2 |
| 37 | 1+(−4.57+2.64i)T+(684.5−1.18e3i)T2 |
| 41 | 1+(−12.4−21.6i)T+(−840.5+1.45e3i)T2 |
| 43 | 1+(0.388−0.673i)T+(−924.5−1.60e3i)T2 |
| 47 | 1+29.1T+2.20e3T2 |
| 53 | 1+51.3iT−2.80e3T2 |
| 59 | 1+(−46.0+79.7i)T+(−1.74e3−3.01e3i)T2 |
| 61 | 1+(31.3−54.3i)T+(−1.86e3−3.22e3i)T2 |
| 67 | 1+(−7.81+4.51i)T+(2.24e3−3.88e3i)T2 |
| 71 | 1+(38.3−66.3i)T+(−2.52e3−4.36e3i)T2 |
| 73 | 1+124.iT−5.32e3T2 |
| 79 | 1−0.898T+6.24e3T2 |
| 83 | 1+97.0T+6.88e3T2 |
| 89 | 1+(8.09+14.0i)T+(−3.96e3+6.85e3i)T2 |
| 97 | 1+(105.+60.7i)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
−12.87274315200476836035541756130, −11.58698874056072727830534860649, −10.90356676302431109027319727180, −9.855783240187869837447705060516, −8.682237381887945389640676090448, −7.06788496404960782339487716615, −6.44548920292800620766722815649, −4.52079737144504193092315420511, −3.08619383568959488511325257194, −0.52867536125554189063262702114,
2.97551198317308714586116462692, 4.00226100904252037948458340888, 6.19358572118670195791432460730, 7.07504324903172577713107445799, 8.399728792599686562807687417764, 9.057926203218429840487862690279, 10.89242043968348870371331500779, 11.67403674589327366640537763752, 12.64273803336084361453882823953, 13.31949874421345704013405444061