L(s) = 1 | + (0.923 − 0.382i)3-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + 13-s + (−0.707 − 0.707i)19-s − i·21-s + (0.923 + 0.382i)23-s + (0.382 − 0.923i)27-s + (0.923 − 0.382i)29-s + (0.382 + 0.923i)31-s + i·33-s + (−0.923 + 0.382i)37-s + (0.923 − 0.382i)39-s + (−0.923 − 0.382i)41-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)3-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + 13-s + (−0.707 − 0.707i)19-s − i·21-s + (0.923 + 0.382i)23-s + (0.382 − 0.923i)27-s + (0.923 − 0.382i)29-s + (0.382 + 0.923i)31-s + i·33-s + (−0.923 + 0.382i)37-s + (0.923 − 0.382i)39-s + (−0.923 − 0.382i)41-s + ⋯ |
Λ(s)=(=(680s/2ΓR(s)L(s)(0.657−0.753i)Λ(1−s)
Λ(s)=(=(680s/2ΓR(s)L(s)(0.657−0.753i)Λ(1−s)
Degree: |
1 |
Conductor: |
680
= 23⋅5⋅17
|
Sign: |
0.657−0.753i
|
Analytic conductor: |
3.15790 |
Root analytic conductor: |
3.15790 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ680(517,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 680, (0: ), 0.657−0.753i)
|
Particular Values
L(21) |
≈ |
1.951745924−0.8864848242i |
L(21) |
≈ |
1.951745924−0.8864848242i |
L(1) |
≈ |
1.498235718−0.3593236479i |
L(1) |
≈ |
1.498235718−0.3593236479i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
| 17 | 1 |
good | 3 | 1+(0.923−0.382i)T |
| 7 | 1+(0.382−0.923i)T |
| 11 | 1+(−0.382+0.923i)T |
| 13 | 1+T |
| 19 | 1+(−0.707−0.707i)T |
| 23 | 1+(0.923+0.382i)T |
| 29 | 1+(0.923−0.382i)T |
| 31 | 1+(0.382+0.923i)T |
| 37 | 1+(−0.923+0.382i)T |
| 41 | 1+(−0.923−0.382i)T |
| 43 | 1+(−0.707−0.707i)T |
| 47 | 1+T |
| 53 | 1+(0.707−0.707i)T |
| 59 | 1+(0.707−0.707i)T |
| 61 | 1+(−0.923−0.382i)T |
| 67 | 1−iT |
| 71 | 1+(−0.382−0.923i)T |
| 73 | 1+(0.382+0.923i)T |
| 79 | 1+(−0.382+0.923i)T |
| 83 | 1+(−0.707+0.707i)T |
| 89 | 1−iT |
| 97 | 1+(0.382+0.923i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.770321424930024801300846152167, −21.68492671126937770332101346703, −21.19048955434083578445313873025, −20.63629497396399881852087479756, −19.505403526336452651538410428111, −18.72548023839555976920011503088, −18.32916683937249851731173831920, −16.92684994438717084128747288591, −16.06176205203236332587900222368, −15.35837627772345928994639399801, −14.660219850226667699098023845871, −13.74674426833237312651837676137, −13.06353876700088067326685890187, −11.98560837513914083331029675154, −10.92357833573515209639195777648, −10.25202023207020860647481967577, −8.91068083464219236701274220265, −8.613237750456424285855139771493, −7.79863829012998536068595057763, −6.42643610022688948841898380715, −5.484914201153367834124953849527, −4.44361347217686753231951839811, −3.35875915754877970916389640691, −2.574769907689064180883670249695, −1.43424075172300046559203932159,
1.07220258290206145248102867384, 2.04791405354760605270277428563, 3.213959446321922453163615488349, 4.14052023482100091183224689898, 5.058001619288524828857361239997, 6.71431740028138240021190394841, 7.112058381196982128101130776973, 8.23409812608486022980792998080, 8.80675958108173701169453582766, 10.01761218283718099739918216501, 10.66366989510964171981645468550, 11.83383190049601272896880030380, 12.88626500680980720628977264642, 13.52365363212984159930960599570, 14.17961758269109634602043933468, 15.227795115952633790475808879975, 15.711870494419968076267772513354, 17.106372016552229481276850150495, 17.72008266387827692615502131445, 18.60603400009921021416773973977, 19.4705519671999203450648122845, 20.20983564496752834481033332973, 20.85663687909638034435219035888, 21.465263654830789679100661040619, 22.93130952895219350388712913967