L(s) = 1 | + (0.382 − 0.923i)2-s + (0.706 + 0.140i)3-s + (−0.707 − 0.707i)4-s + (0.400 − 0.598i)6-s + (−1.70 + 2.55i)7-s + (−0.923 + 0.382i)8-s + (−2.29 − 0.949i)9-s + (−1.25 − 0.838i)11-s + (−0.400 − 0.598i)12-s − 5.77·13-s + (1.70 + 2.55i)14-s + i·16-s + (0.809 + 4.04i)17-s + (−1.75 + 1.75i)18-s + (−1.58 + 0.654i)19-s + ⋯ |
L(s) = 1 | + (0.270 − 0.653i)2-s + (0.407 + 0.0811i)3-s + (−0.353 − 0.353i)4-s + (0.163 − 0.244i)6-s + (−0.645 + 0.965i)7-s + (−0.326 + 0.135i)8-s + (−0.764 − 0.316i)9-s + (−0.378 − 0.252i)11-s + (−0.115 − 0.172i)12-s − 1.60·13-s + (0.456 + 0.682i)14-s + 0.250i·16-s + (0.196 + 0.980i)17-s + (−0.413 + 0.413i)18-s + (−0.362 + 0.150i)19-s + ⋯ |
Λ(s)=(=(850s/2ΓC(s)L(s)(−0.611−0.791i)Λ(2−s)
Λ(s)=(=(850s/2ΓC(s+1/2)L(s)(−0.611−0.791i)Λ(1−s)
Degree: |
2 |
Conductor: |
850
= 2⋅52⋅17
|
Sign: |
−0.611−0.791i
|
Analytic conductor: |
6.78728 |
Root analytic conductor: |
2.60524 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ850(107,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 850, ( :1/2), −0.611−0.791i)
|
Particular Values
L(1) |
≈ |
0.0973083+0.198081i |
L(21) |
≈ |
0.0973083+0.198081i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.382+0.923i)T |
| 5 | 1 |
| 17 | 1+(−0.809−4.04i)T |
good | 3 | 1+(−0.706−0.140i)T+(2.77+1.14i)T2 |
| 7 | 1+(1.70−2.55i)T+(−2.67−6.46i)T2 |
| 11 | 1+(1.25+0.838i)T+(4.20+10.1i)T2 |
| 13 | 1+5.77T+13T2 |
| 19 | 1+(1.58−0.654i)T+(13.4−13.4i)T2 |
| 23 | 1+(0.204+1.02i)T+(−21.2+8.80i)T2 |
| 29 | 1+(−0.749+3.77i)T+(−26.7−11.0i)T2 |
| 31 | 1+(3.73−2.49i)T+(11.8−28.6i)T2 |
| 37 | 1+(1.49−7.53i)T+(−34.1−14.1i)T2 |
| 41 | 1+(0.447+2.24i)T+(−37.8+15.6i)T2 |
| 43 | 1+(2.77+6.69i)T+(−30.4+30.4i)T2 |
| 47 | 1+10.8iT−47T2 |
| 53 | 1+(−2.42−1.00i)T+(37.4+37.4i)T2 |
| 59 | 1+(2.59−6.25i)T+(−41.7−41.7i)T2 |
| 61 | 1+(−14.3+2.85i)T+(56.3−23.3i)T2 |
| 67 | 1+(11.0−11.0i)T−67iT2 |
| 71 | 1+(−6.97−10.4i)T+(−27.1+65.5i)T2 |
| 73 | 1+(1.30+1.94i)T+(−27.9+67.4i)T2 |
| 79 | 1+(−1.89+2.83i)T+(−30.2−72.9i)T2 |
| 83 | 1+(5.71−13.8i)T+(−58.6−58.6i)T2 |
| 89 | 1+(3.10+3.10i)T+89iT2 |
| 97 | 1+(8.08+12.0i)T+(−37.1+89.6i)T2 |
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
−10.26274079427975618212183657402, −9.828617517622865340611695585314, −8.785701559462586167957282709509, −8.347009852533950547961643077165, −6.98953939054678010329650377076, −5.87325841026573072702663714130, −5.24326431723622741524583071016, −3.90619211362283742510446966101, −2.89116104674407369111018638529, −2.20518518908869782233547357869,
0.083966297925478681888963990025, 2.46105042268659839809230124973, 3.40736262928865303555919204234, 4.64076888237650213177880223718, 5.38775107755300907071429681580, 6.59097174877960329937886555231, 7.44633727489165210746655889178, 7.80996723237701475209249850332, 9.099586478221683074601667113425, 9.675451994678620082001791745307