L(s) = 1 | + (−0.494 − 1.84i)2-s + (−1.43 + 0.826i)4-s + (0.541 − 2.16i)5-s + (2.64 − 0.0471i)7-s + (−0.468 − 0.468i)8-s + (−4.27 + 0.0725i)10-s + (0.971 − 0.561i)11-s + (0.830 − 0.830i)13-s + (−1.39 − 4.86i)14-s + (−2.28 + 3.96i)16-s + (2.16 + 0.578i)17-s + (−5.19 − 3.00i)19-s + (1.01 + 3.55i)20-s + (−1.51 − 1.51i)22-s + (0.641 − 0.171i)23-s + ⋯ |
L(s) = 1 | + (−0.349 − 1.30i)2-s + (−0.715 + 0.413i)4-s + (0.242 − 0.970i)5-s + (0.999 − 0.0178i)7-s + (−0.165 − 0.165i)8-s + (−1.35 + 0.0229i)10-s + (0.293 − 0.169i)11-s + (0.230 − 0.230i)13-s + (−0.372 − 1.29i)14-s + (−0.571 + 0.990i)16-s + (0.523 + 0.140i)17-s + (−1.19 − 0.688i)19-s + (0.227 + 0.794i)20-s + (−0.323 − 0.323i)22-s + (0.133 − 0.0358i)23-s + ⋯ |
Λ(s)=(=(315s/2ΓC(s)L(s)(−0.886+0.463i)Λ(2−s)
Λ(s)=(=(315s/2ΓC(s+1/2)L(s)(−0.886+0.463i)Λ(1−s)
Degree: |
2 |
Conductor: |
315
= 32⋅5⋅7
|
Sign: |
−0.886+0.463i
|
Analytic conductor: |
2.51528 |
Root analytic conductor: |
1.58596 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ315(107,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 315, ( :1/2), −0.886+0.463i)
|
Particular Values
L(1) |
≈ |
0.285191−1.16080i |
L(21) |
≈ |
0.285191−1.16080i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1+(−0.541+2.16i)T |
| 7 | 1+(−2.64+0.0471i)T |
good | 2 | 1+(0.494+1.84i)T+(−1.73+i)T2 |
| 11 | 1+(−0.971+0.561i)T+(5.5−9.52i)T2 |
| 13 | 1+(−0.830+0.830i)T−13iT2 |
| 17 | 1+(−2.16−0.578i)T+(14.7+8.5i)T2 |
| 19 | 1+(5.19+3.00i)T+(9.5+16.4i)T2 |
| 23 | 1+(−0.641+0.171i)T+(19.9−11.5i)T2 |
| 29 | 1+9.88T+29T2 |
| 31 | 1+(−4.19−7.27i)T+(−15.5+26.8i)T2 |
| 37 | 1+(−5.51+1.47i)T+(32.0−18.5i)T2 |
| 41 | 1−3.31iT−41T2 |
| 43 | 1+(−5.26+5.26i)T−43iT2 |
| 47 | 1+(−2.55−9.52i)T+(−40.7+23.5i)T2 |
| 53 | 1+(−0.211+0.790i)T+(−45.8−26.5i)T2 |
| 59 | 1+(−1.75−3.03i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−2.57+4.46i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−1.66+6.20i)T+(−58.0−33.5i)T2 |
| 71 | 1+12.1iT−71T2 |
| 73 | 1+(4.49+1.20i)T+(63.2+36.5i)T2 |
| 79 | 1+(−9.12−5.27i)T+(39.5+68.4i)T2 |
| 83 | 1+(−11.5−11.5i)T+83iT2 |
| 89 | 1+(−1.76+3.06i)T+(−44.5−77.0i)T2 |
| 97 | 1+(−10.6−10.6i)T+97iT2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.15846198217220405695349501669, −10.59455250385063395183548124932, −9.390007724802954480180092245368, −8.798225079525620101200731414217, −7.83524308110510254191795979912, −6.19235908373438282496738322380, −4.92643389300017796479781231262, −3.85193269002316705458202892584, −2.21594261549918780028833326637, −1.05890149479298471405614960692,
2.22950863884669519860102337695, 4.06841232542458750119293363683, 5.55366283543383302793625847404, 6.29192161548858702862771008973, 7.34335846279774840745680444996, 7.958752769378978805605994986451, 8.984432142111374788677125319955, 10.02855516337740173652769231611, 11.14145976499892679973716520663, 11.78904760187755004333681031151