L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 2·7-s − 8-s + 10-s − 10·11-s − 5·13-s − 2·14-s + 15-s − 16-s + 5·17-s − 2·19-s − 2·21-s − 10·22-s − 24-s − 5·26-s − 27-s − 18·29-s + 30-s + 8·31-s − 10·33-s + 5·34-s − 2·35-s + 37-s − 2·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 3.01·11-s − 1.38·13-s − 0.534·14-s + 0.258·15-s − 1/4·16-s + 1.21·17-s − 0.458·19-s − 0.436·21-s − 2.13·22-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 3.34·29-s + 0.182·30-s + 1.43·31-s − 1.74·33-s + 0.857·34-s − 0.338·35-s + 0.164·37-s − 0.324·38-s + ⋯ |
Λ(s)=(=(1232100s/2ΓC(s)2L(s)Λ(2−s)
Λ(s)=(=(1232100s/2ΓC(s+1/2)2L(s)Λ(1−s)
Degree: |
4 |
Conductor: |
1232100
= 22⋅32⋅52⋅372
|
Sign: |
1
|
Analytic conductor: |
78.5597 |
Root analytic conductor: |
2.97714 |
Motivic weight: |
1 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(4, 1232100, ( :1/2,1/2), 1)
|
Particular Values
L(1) |
≈ |
1.132181111 |
L(21) |
≈ |
1.132181111 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C2 | 1−T+T2 |
| 3 | C2 | 1−T+T2 |
| 5 | C2 | 1−T+T2 |
| 37 | C2 | 1−T+pT2 |
good | 7 | C22 | 1+2T−3T2+2pT3+p2T4 |
| 11 | C2 | (1+5T+pT2)2 |
| 13 | C2 | (1−2T+pT2)(1+7T+pT2) |
| 17 | C22 | 1−5T+8T2−5pT3+p2T4 |
| 19 | C22 | 1+2T−15T2+2pT3+p2T4 |
| 23 | C2 | (1+pT2)2 |
| 29 | C2 | (1+9T+pT2)2 |
| 31 | C2 | (1−4T+pT2)2 |
| 41 | C22 | 1+6T−5T2+6pT3+p2T4 |
| 43 | C2 | (1−4T+pT2)2 |
| 47 | C2 | (1−9T+pT2)2 |
| 53 | C22 | 1+6T−17T2+6pT3+p2T4 |
| 59 | C22 | 1+3T−50T2+3pT3+p2T4 |
| 61 | C22 | 1−pT2+p2T4 |
| 67 | C22 | 1−9T+14T2−9pT3+p2T4 |
| 71 | C22 | 1−12T+73T2−12pT3+p2T4 |
| 73 | C2 | (1+12T+pT2)2 |
| 79 | C22 | 1+16T+177T2+16pT3+p2T4 |
| 83 | C22 | 1−2T−79T2−2pT3+p2T4 |
| 89 | C22 | 1−14T+107T2−14pT3+p2T4 |
| 97 | C2 | (1+18T+pT2)2 |
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L(s)=p∏ j=1∏4(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.895229605154386705753371611054, −9.830617483174062762974913241447, −9.258885623341698062529608626560, −8.952677386355150180209916458371, −8.279524988625482730407823827717, −7.79388202743721958840015857397, −7.67797784157869321364174352747, −7.27202117645734074763766748881, −6.79331068334586798801399775171, −5.92412104781530460098390377473, −5.67999674593804376707799382122, −5.25889641366113415656529714713, −5.22187997594200934365590020570, −4.17533733727464206227780713276, −4.07700134980005773011811315758, −3.01787900420208888078971942426, −2.91216122600353537452807139657, −2.46533774132156703503218947021, −1.90660766179344916812031405289, −0.36968190955454456474643329672,
0.36968190955454456474643329672, 1.90660766179344916812031405289, 2.46533774132156703503218947021, 2.91216122600353537452807139657, 3.01787900420208888078971942426, 4.07700134980005773011811315758, 4.17533733727464206227780713276, 5.22187997594200934365590020570, 5.25889641366113415656529714713, 5.67999674593804376707799382122, 5.92412104781530460098390377473, 6.79331068334586798801399775171, 7.27202117645734074763766748881, 7.67797784157869321364174352747, 7.79388202743721958840015857397, 8.279524988625482730407823827717, 8.952677386355150180209916458371, 9.258885623341698062529608626560, 9.830617483174062762974913241447, 9.895229605154386705753371611054