L(s) = 1 | + (−1.5 − 0.866i)3-s + (−1 + 1.73i)4-s + (3 + 1.73i)5-s + (3 − 1.73i)7-s + (1.5 + 2.59i)9-s + (−3 + 1.73i)11-s + (3 − 1.73i)12-s + (3.5 + 0.866i)13-s + (−3 − 5.19i)15-s + (−1.99 − 3.46i)16-s − 3·17-s − 3.46i·19-s + (−6 + 3.46i)20-s − 6·21-s + (−1.5 + 2.59i)23-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (−0.5 + 0.866i)4-s + (1.34 + 0.774i)5-s + (1.13 − 0.654i)7-s + (0.5 + 0.866i)9-s + (−0.904 + 0.522i)11-s + (0.866 − 0.499i)12-s + (0.970 + 0.240i)13-s + (−0.774 − 1.34i)15-s + (−0.499 − 0.866i)16-s − 0.727·17-s − 0.794i·19-s + (−1.34 + 0.774i)20-s − 1.30·21-s + (−0.312 + 0.541i)23-s + ⋯ |
Λ(s)=(=(117s/2ΓC(s)L(s)(0.898−0.439i)Λ(2−s)
Λ(s)=(=(117s/2ΓC(s+1/2)L(s)(0.898−0.439i)Λ(1−s)
Degree: |
2 |
Conductor: |
117
= 32⋅13
|
Sign: |
0.898−0.439i
|
Analytic conductor: |
0.934249 |
Root analytic conductor: |
0.966565 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ117(103,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 117, ( :1/2), 0.898−0.439i)
|
Particular Values
L(1) |
≈ |
0.930548+0.215707i |
L(21) |
≈ |
0.930548+0.215707i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.5+0.866i)T |
| 13 | 1+(−3.5−0.866i)T |
good | 2 | 1+(1−1.73i)T2 |
| 5 | 1+(−3−1.73i)T+(2.5+4.33i)T2 |
| 7 | 1+(−3+1.73i)T+(3.5−6.06i)T2 |
| 11 | 1+(3−1.73i)T+(5.5−9.52i)T2 |
| 17 | 1+3T+17T2 |
| 19 | 1+3.46iT−19T2 |
| 23 | 1+(1.5−2.59i)T+(−11.5−19.9i)T2 |
| 29 | 1+(3+5.19i)T+(−14.5+25.1i)T2 |
| 31 | 1+(3+1.73i)T+(15.5+26.8i)T2 |
| 37 | 1+6.92iT−37T2 |
| 41 | 1+(−6−3.46i)T+(20.5+35.5i)T2 |
| 43 | 1+(−0.5−0.866i)T+(−21.5+37.2i)T2 |
| 47 | 1+(−6+3.46i)T+(23.5−40.7i)T2 |
| 53 | 1+9T+53T2 |
| 59 | 1+(3+1.73i)T+(29.5+51.0i)T2 |
| 61 | 1+(3.5+6.06i)T+(−30.5+52.8i)T2 |
| 67 | 1+(33.5+58.0i)T2 |
| 71 | 1−71T2 |
| 73 | 1−10.3iT−73T2 |
| 79 | 1+(0.5+0.866i)T+(−39.5+68.4i)T2 |
| 83 | 1+(6−3.46i)T+(41.5−71.8i)T2 |
| 89 | 1+3.46iT−89T2 |
| 97 | 1+(6−3.46i)T+(48.5−84.0i)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
−13.49569157970577363403370950556, −12.86317277508308558527064176802, −11.30218504250711364717709228963, −10.79949453063832226856875051186, −9.485335105771080245116572571383, −7.926302299218360394745824417957, −7.04529436021832103989945153038, −5.71602598961146143900982904690, −4.45005616763899648969067025496, −2.14542912532483567094193497700,
1.54224651653089735279217434710, 4.66233783353132560538996593963, 5.53403650498432787604332572436, 6.03034423006051940481737598954, 8.511129000410489019414212713400, 9.270092783501581373863127419904, 10.45949931341699129739439165294, 11.02991091580983695638037644385, 12.53437027023792065317927091281, 13.47508031199557530117133950960