L(s) = 1 | + (−0.0729 + 1.39i)2-s + (2.18 − 1.77i)3-s + (0.0566 + 0.00595i)4-s + (−0.287 + 2.21i)5-s + (2.30 + 3.17i)6-s + (−2.62 − 0.311i)7-s + (−0.448 + 2.83i)8-s + (1.02 − 4.82i)9-s + (−3.06 − 0.562i)10-s + (1.26 − 0.268i)11-s + (0.134 − 0.0873i)12-s + (1.23 − 2.42i)13-s + (0.624 − 3.63i)14-s + (3.30 + 5.36i)15-s + (−3.79 − 0.807i)16-s + (1.66 − 0.639i)17-s + ⋯ |
L(s) = 1 | + (−0.0515 + 0.984i)2-s + (1.26 − 1.02i)3-s + (0.0283 + 0.00297i)4-s + (−0.128 + 0.991i)5-s + (0.942 + 1.29i)6-s + (−0.993 − 0.117i)7-s + (−0.158 + 1.00i)8-s + (0.341 − 1.60i)9-s + (−0.969 − 0.177i)10-s + (0.380 − 0.0808i)11-s + (0.0388 − 0.0252i)12-s + (0.342 − 0.672i)13-s + (0.167 − 0.971i)14-s + (0.852 + 1.38i)15-s + (−0.949 − 0.201i)16-s + (0.403 − 0.155i)17-s + ⋯ |
Λ(s)=(=(175s/2ΓC(s)L(s)(0.699−0.714i)Λ(2−s)
Λ(s)=(=(175s/2ΓC(s+1/2)L(s)(0.699−0.714i)Λ(1−s)
Degree: |
2 |
Conductor: |
175
= 52⋅7
|
Sign: |
0.699−0.714i
|
Analytic conductor: |
1.39738 |
Root analytic conductor: |
1.18210 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ175(47,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 175, ( :1/2), 0.699−0.714i)
|
Particular Values
L(1) |
≈ |
1.47935+0.621663i |
L(21) |
≈ |
1.47935+0.621663i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(0.287−2.21i)T |
| 7 | 1+(2.62+0.311i)T |
good | 2 | 1+(0.0729−1.39i)T+(−1.98−0.209i)T2 |
| 3 | 1+(−2.18+1.77i)T+(0.623−2.93i)T2 |
| 11 | 1+(−1.26+0.268i)T+(10.0−4.47i)T2 |
| 13 | 1+(−1.23+2.42i)T+(−7.64−10.5i)T2 |
| 17 | 1+(−1.66+0.639i)T+(12.6−11.3i)T2 |
| 19 | 1+(0.525+4.99i)T+(−18.5+3.95i)T2 |
| 23 | 1+(7.99+0.419i)T+(22.8+2.40i)T2 |
| 29 | 1+(−0.927+1.27i)T+(−8.96−27.5i)T2 |
| 31 | 1+(3.58−8.05i)T+(−20.7−23.0i)T2 |
| 37 | 1+(1.62+2.50i)T+(−15.0+33.8i)T2 |
| 41 | 1+(−11.6+3.78i)T+(33.1−24.0i)T2 |
| 43 | 1+(4.19−4.19i)T−43iT2 |
| 47 | 1+(2.18−5.68i)T+(−34.9−31.4i)T2 |
| 53 | 1+(2.51+3.10i)T+(−11.0+51.8i)T2 |
| 59 | 1+(2.98−3.31i)T+(−6.16−58.6i)T2 |
| 61 | 1+(−6.76+6.08i)T+(6.37−60.6i)T2 |
| 67 | 1+(−1.57−4.11i)T+(−49.7+44.8i)T2 |
| 71 | 1+(−6.77−4.91i)T+(21.9+67.5i)T2 |
| 73 | 1+(3.04+1.97i)T+(29.6+66.6i)T2 |
| 79 | 1+(1.50+3.38i)T+(−52.8+58.7i)T2 |
| 83 | 1+(−0.0807−0.0127i)T+(78.9+25.6i)T2 |
| 89 | 1+(−10.7−11.9i)T+(−9.30+88.5i)T2 |
| 97 | 1+(−5.33+0.844i)T+(92.2−29.9i)T2 |
show more | |
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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
−13.14388624911349000937303895287, −12.12888035413503956771202331496, −10.81761015851968560428832156913, −9.472515163133284369208131138043, −8.358100944320597604258083286509, −7.50781966016923249028267675469, −6.80429568534588143541461286925, −6.03669360331535512403184403425, −3.44577640357779218293343461334, −2.45476617106814728836594947615,
2.03410171525777927263358635214, 3.59856069422125932089156893034, 4.10988781139594189266387563211, 6.07865377969781415250386143307, 7.87060022859823523963572984856, 8.993647666307694606127455277454, 9.670866678859924264192571862619, 10.22882409664185781641857225674, 11.67136484487784962927691246068, 12.50858218549619572768439201253