L(s) = 1 | + (0.148 + 0.107i)2-s + (−0.454 − 1.39i)3-s + (−0.607 − 1.87i)4-s + (0.0833 − 0.256i)6-s − 3.26·7-s + (0.224 − 0.690i)8-s + (0.674 − 0.489i)9-s + (−1.61 − 1.17i)11-s + (−2.34 + 1.70i)12-s + (0.239 − 0.174i)13-s + (−0.483 − 0.351i)14-s + (−3.07 + 2.23i)16-s + (−1.59 + 4.91i)17-s + 0.152·18-s + (0.534 − 1.64i)19-s + ⋯ |
L(s) = 1 | + (0.104 + 0.0761i)2-s + (−0.262 − 0.808i)3-s + (−0.303 − 0.935i)4-s + (0.0340 − 0.104i)6-s − 1.23·7-s + (0.0793 − 0.244i)8-s + (0.224 − 0.163i)9-s + (−0.487 − 0.354i)11-s + (−0.675 + 0.491i)12-s + (0.0665 − 0.0483i)13-s + (−0.129 − 0.0938i)14-s + (−0.768 + 0.558i)16-s + (−0.386 + 1.19i)17-s + 0.0359·18-s + (0.122 − 0.377i)19-s + ⋯ |
Λ(s)=(=(625s/2ΓC(s)L(s)(−0.904−0.425i)Λ(2−s)
Λ(s)=(=(625s/2ΓC(s+1/2)L(s)(−0.904−0.425i)Λ(1−s)
Degree: |
2 |
Conductor: |
625
= 54
|
Sign: |
−0.904−0.425i
|
Analytic conductor: |
4.99065 |
Root analytic conductor: |
2.23397 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ625(376,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 625, ( :1/2), −0.904−0.425i)
|
Particular Values
L(1) |
≈ |
0.103595+0.463461i |
L(21) |
≈ |
0.103595+0.463461i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
good | 2 | 1+(−0.148−0.107i)T+(0.618+1.90i)T2 |
| 3 | 1+(0.454+1.39i)T+(−2.42+1.76i)T2 |
| 7 | 1+3.26T+7T2 |
| 11 | 1+(1.61+1.17i)T+(3.39+10.4i)T2 |
| 13 | 1+(−0.239+0.174i)T+(4.01−12.3i)T2 |
| 17 | 1+(1.59−4.91i)T+(−13.7−9.99i)T2 |
| 19 | 1+(−0.534+1.64i)T+(−15.3−11.1i)T2 |
| 23 | 1+(0.711+0.516i)T+(7.10+21.8i)T2 |
| 29 | 1+(−1.82−5.62i)T+(−23.4+17.0i)T2 |
| 31 | 1+(−1.88+5.80i)T+(−25.0−18.2i)T2 |
| 37 | 1+(6.54−4.75i)T+(11.4−35.1i)T2 |
| 41 | 1+(−0.821+0.596i)T+(12.6−38.9i)T2 |
| 43 | 1+3.24T+43T2 |
| 47 | 1+(−1.30−4.01i)T+(−38.0+27.6i)T2 |
| 53 | 1+(2.50+7.70i)T+(−42.8+31.1i)T2 |
| 59 | 1+(4.80−3.48i)T+(18.2−56.1i)T2 |
| 61 | 1+(0.740+0.538i)T+(18.8+58.0i)T2 |
| 67 | 1+(−2.12+6.55i)T+(−54.2−39.3i)T2 |
| 71 | 1+(1.84+5.67i)T+(−57.4+41.7i)T2 |
| 73 | 1+(7.14+5.19i)T+(22.5+69.4i)T2 |
| 79 | 1+(2.39+7.38i)T+(−63.9+46.4i)T2 |
| 83 | 1+(−4.48+13.8i)T+(−67.1−48.7i)T2 |
| 89 | 1+(6.08+4.42i)T+(27.5+84.6i)T2 |
| 97 | 1+(2.07+6.39i)T+(−78.4+57.0i)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
−10.14682520090661308623748955134, −9.356804459501565880500164725572, −8.397876649364017300644094738019, −7.14526713685566798578712612110, −6.35875046747294065846334799433, −5.89768711700079069403823385074, −4.60461972244240874854870165524, −3.31540511669543602159958471831, −1.70981889044240014554306706027, −0.25361856198140127864437841069,
2.60970622539301118039507228355, 3.60842975345267817801450237214, 4.50069349865512956302245020122, 5.40532754456736821177181456974, 6.77359939344280756447750777841, 7.51189386641160859246334040002, 8.639718826916471985913184351634, 9.559194524379681567579814063267, 10.05467683582807542353620638071, 11.04383095119051804040265730676