L(s) = 1 | + (0.965 − 0.258i)2-s + (0.161 + 1.72i)3-s + (0.866 − 0.499i)4-s + (−2.21 + 0.593i)5-s + (0.602 + 1.62i)6-s + (2.94 + 2.94i)7-s + (0.707 − 0.707i)8-s + (−2.94 + 0.556i)9-s + (−1.98 + 1.14i)10-s + (−0.431 − 1.60i)11-s + (1.00 + 1.41i)12-s + (1.58 + 3.23i)13-s + (3.60 + 2.08i)14-s + (−1.38 − 3.72i)15-s + (0.500 − 0.866i)16-s + (1.96 − 3.41i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.0932 + 0.995i)3-s + (0.433 − 0.249i)4-s + (−0.990 + 0.265i)5-s + (0.245 + 0.662i)6-s + (1.11 + 1.11i)7-s + (0.249 − 0.249i)8-s + (−0.982 + 0.185i)9-s + (−0.628 + 0.362i)10-s + (−0.129 − 0.485i)11-s + (0.289 + 0.407i)12-s + (0.439 + 0.898i)13-s + (0.964 + 0.556i)14-s + (−0.356 − 0.961i)15-s + (0.125 − 0.216i)16-s + (0.477 − 0.827i)17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(0.495−0.868i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(0.495−0.868i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
0.495−0.868i
|
Analytic conductor: |
1.86849 |
Root analytic conductor: |
1.36693 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ234(41,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 234, ( :1/2), 0.495−0.868i)
|
Particular Values
L(1) |
≈ |
1.47800+0.858484i |
L(21) |
≈ |
1.47800+0.858484i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.965+0.258i)T |
| 3 | 1+(−0.161−1.72i)T |
| 13 | 1+(−1.58−3.23i)T |
good | 5 | 1+(2.21−0.593i)T+(4.33−2.5i)T2 |
| 7 | 1+(−2.94−2.94i)T+7iT2 |
| 11 | 1+(0.431+1.60i)T+(−9.52+5.5i)T2 |
| 17 | 1+(−1.96+3.41i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.878+3.27i)T+(−16.4+9.5i)T2 |
| 23 | 1−5.07T+23T2 |
| 29 | 1+(4.53+2.61i)T+(14.5+25.1i)T2 |
| 31 | 1+(0.610+2.27i)T+(−26.8+15.5i)T2 |
| 37 | 1+(−1.21+4.55i)T+(−32.0−18.5i)T2 |
| 41 | 1+(−2.88−2.88i)T+41iT2 |
| 43 | 1+4.92iT−43T2 |
| 47 | 1+(−4.39−1.17i)T+(40.7+23.5i)T2 |
| 53 | 1+6.69iT−53T2 |
| 59 | 1+(8.36+2.24i)T+(51.0+29.5i)T2 |
| 61 | 1−11.6T+61T2 |
| 67 | 1+(9.81−9.81i)T−67iT2 |
| 71 | 1+(13.5−3.63i)T+(61.4−35.5i)T2 |
| 73 | 1+(−6.61−6.61i)T+73iT2 |
| 79 | 1+(8.13+14.0i)T+(−39.5+68.4i)T2 |
| 83 | 1+(2.02−7.55i)T+(−71.8−41.5i)T2 |
| 89 | 1+(−8.95−2.40i)T+(77.0+44.5i)T2 |
| 97 | 1+(6.05−6.05i)T−97iT2 |
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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
−11.87468029941098504461278393073, −11.42877164679856774867327121856, −10.92484104372173002773331024154, −9.341779654041291949835848556995, −8.555422650214004070370147074693, −7.40550295390405902315908570520, −5.77098102103636828503764268694, −4.86621769724656660292531880242, −3.85997956113235884820837962234, −2.58370289961841285962198997534,
1.38116461148379341500836611215, 3.42564655659896838692335931939, 4.55837439515377923400441498286, 5.82843035570733005325632897713, 7.28108643884573526483019671867, 7.75597279842494361815538367693, 8.502753015255585506027179553604, 10.56094736053001621615739777649, 11.25197850467177130254035402577, 12.25630363429544012397180794921