L(s) = 1 | + (0.707 − 0.707i)2-s + (1.72 + 0.0878i)3-s − 1.00i·4-s + (−2.64 − 0.707i)5-s + (1.28 − 1.16i)6-s + (3.36 + 0.902i)7-s + (−0.707 − 0.707i)8-s + (2.98 + 0.303i)9-s + (−2.36 + 1.36i)10-s + (−1.29 − 1.29i)11-s + (0.0878 − 1.72i)12-s + (2.77 − 2.29i)13-s + (3.01 − 1.74i)14-s + (−4.50 − 1.45i)15-s − 1.00·16-s + (−0.419 + 0.726i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.998 + 0.0507i)3-s − 0.500i·4-s + (−1.18 − 0.316i)5-s + (0.524 − 0.473i)6-s + (1.27 + 0.340i)7-s + (−0.250 − 0.250i)8-s + (0.994 + 0.101i)9-s + (−0.748 + 0.432i)10-s + (−0.389 − 0.389i)11-s + (0.0253 − 0.499i)12-s + (0.770 − 0.637i)13-s + (0.806 − 0.465i)14-s + (−1.16 − 0.375i)15-s − 0.250·16-s + (−0.101 + 0.176i)17-s + ⋯ |
Λ(s)=(=(234s/2ΓC(s)L(s)(0.656+0.754i)Λ(2−s)
Λ(s)=(=(234s/2ΓC(s+1/2)L(s)(0.656+0.754i)Λ(1−s)
Degree: |
2 |
Conductor: |
234
= 2⋅32⋅13
|
Sign: |
0.656+0.754i
|
Analytic conductor: |
1.86849 |
Root analytic conductor: |
1.36693 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ234(11,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 234, ( :1/2), 0.656+0.754i)
|
Particular Values
L(1) |
≈ |
1.76937−0.806166i |
L(21) |
≈ |
1.76937−0.806166i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.707+0.707i)T |
| 3 | 1+(−1.72−0.0878i)T |
| 13 | 1+(−2.77+2.29i)T |
good | 5 | 1+(2.64+0.707i)T+(4.33+2.5i)T2 |
| 7 | 1+(−3.36−0.902i)T+(6.06+3.5i)T2 |
| 11 | 1+(1.29+1.29i)T+11iT2 |
| 17 | 1+(0.419−0.726i)T+(−8.5−14.7i)T2 |
| 19 | 1+(6.94−1.86i)T+(16.4−9.5i)T2 |
| 23 | 1+(3.18−5.52i)T+(−11.5−19.9i)T2 |
| 29 | 1−6.05iT−29T2 |
| 31 | 1+(1.99−7.43i)T+(−26.8−15.5i)T2 |
| 37 | 1+(4.09+1.09i)T+(32.0+18.5i)T2 |
| 41 | 1+(1.07+4.01i)T+(−35.5+20.5i)T2 |
| 43 | 1+(−7.47+4.31i)T+(21.5−37.2i)T2 |
| 47 | 1+(−7.46+2.00i)T+(40.7−23.5i)T2 |
| 53 | 1+3.92iT−53T2 |
| 59 | 1+(5.26+5.26i)T+59iT2 |
| 61 | 1+(−0.247−0.428i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−5.34+1.43i)T+(58.0−33.5i)T2 |
| 71 | 1+(−1.48−5.55i)T+(−61.4+35.5i)T2 |
| 73 | 1+(9.66−9.66i)T−73iT2 |
| 79 | 1+(0.892−1.54i)T+(−39.5−68.4i)T2 |
| 83 | 1+(4.07+15.2i)T+(−71.8+41.5i)T2 |
| 89 | 1+(−0.385+1.43i)T+(−77.0−44.5i)T2 |
| 97 | 1+(−4.50+16.7i)T+(−84.0−48.5i)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
−12.19280558585957473289216799568, −11.07932063357914598768337077306, −10.45456383442751010771666955922, −8.675579575201664504065788805421, −8.404918584834844251254146394898, −7.34358545763668899932339947467, −5.49623051706310724994070006952, −4.28645555823397683942387694710, −3.44153486310953352846277930990, −1.78011256973782064655196993641,
2.33450289998816887439109543694, 4.14401396106080470960231354727, 4.37569652384319929645317373701, 6.44019503729164085761689284523, 7.62668364499443227069868532478, 8.036682561711370426347950937832, 8.955976325421976565540661973571, 10.59804580090308258033108526103, 11.41124286123069041204551627465, 12.43440631593047181392952151273