L(s) = 1 | + (−0.986 − 0.161i)2-s + (0.425 − 0.905i)3-s + (0.947 + 0.319i)4-s + (0.581 + 0.813i)5-s + (−0.566 + 0.824i)6-s + (0.701 + 0.712i)7-s + (−0.883 − 0.468i)8-s + (−0.638 − 0.769i)9-s + (−0.442 − 0.896i)10-s + (0.241 + 0.970i)11-s + (0.692 − 0.721i)12-s + (−0.961 + 0.276i)13-s + (−0.576 − 0.816i)14-s + (0.983 − 0.181i)15-s + (0.795 + 0.605i)16-s + (−0.107 + 0.994i)17-s + ⋯ |
L(s) = 1 | + (−0.986 − 0.161i)2-s + (0.425 − 0.905i)3-s + (0.947 + 0.319i)4-s + (0.581 + 0.813i)5-s + (−0.566 + 0.824i)6-s + (0.701 + 0.712i)7-s + (−0.883 − 0.468i)8-s + (−0.638 − 0.769i)9-s + (−0.442 − 0.896i)10-s + (0.241 + 0.970i)11-s + (0.692 − 0.721i)12-s + (−0.961 + 0.276i)13-s + (−0.576 − 0.816i)14-s + (0.983 − 0.181i)15-s + (0.795 + 0.605i)16-s + (−0.107 + 0.994i)17-s + ⋯ |
Λ(s)=(=(967s/2ΓR(s)L(s)(−0.0366+0.999i)Λ(1−s)
Λ(s)=(=(967s/2ΓR(s)L(s)(−0.0366+0.999i)Λ(1−s)
Degree: |
1 |
Conductor: |
967
|
Sign: |
−0.0366+0.999i
|
Analytic conductor: |
4.49072 |
Root analytic conductor: |
4.49072 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ967(179,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 967, (0: ), −0.0366+0.999i)
|
Particular Values
L(21) |
≈ |
0.5673300155+0.5885192417i |
L(21) |
≈ |
0.5673300155+0.5885192417i |
L(1) |
≈ |
0.7808924509+0.06885533600i |
L(1) |
≈ |
0.7808924509+0.06885533600i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 967 | 1 |
good | 2 | 1+(−0.986−0.161i)T |
| 3 | 1+(0.425−0.905i)T |
| 5 | 1+(0.581+0.813i)T |
| 7 | 1+(0.701+0.712i)T |
| 11 | 1+(0.241+0.970i)T |
| 13 | 1+(−0.961+0.276i)T |
| 17 | 1+(−0.107+0.994i)T |
| 19 | 1+(−0.927+0.374i)T |
| 23 | 1+(−0.984−0.174i)T |
| 29 | 1+(−0.608−0.793i)T |
| 31 | 1+(−0.465−0.884i)T |
| 37 | 1+(0.483+0.875i)T |
| 41 | 1+(0.682−0.730i)T |
| 43 | 1+(0.254+0.967i)T |
| 47 | 1+(−0.0422+0.999i)T |
| 53 | 1+(−0.247+0.968i)T |
| 59 | 1+(−0.889+0.457i)T |
| 61 | 1+(−0.974−0.225i)T |
| 67 | 1+(0.999−0.0390i)T |
| 71 | 1+(0.353−0.935i)T |
| 73 | 1+(−0.247−0.968i)T |
| 79 | 1+(−0.911−0.410i)T |
| 83 | 1+(0.152−0.988i)T |
| 89 | 1+(−0.533−0.845i)T |
| 97 | 1+(0.623+0.781i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.50669125149848746768657417886, −20.516112938196375465253234980618, −20.03556936258931140560980641643, −19.4840873799564153556146763836, −18.20622560820780923855054525232, −17.455682775556939963431446482041, −16.70975669906785365366765330213, −16.35975431024418154953051293438, −15.4177280857491492099478863016, −14.34576948341728887906549673720, −14.01434404035271983297833554841, −12.72184062084601399315979836519, −11.51306625357022111413807783013, −10.84785688754308496940060377049, −10.0479595088366263912648759117, −9.33141479612008227609706780229, −8.631819902777221100819110121412, −7.96664901674456330065567673814, −6.97554030766417845796796622909, −5.6475428250612234223792761145, −5.0292597349267930510552672779, −3.94389415109873394120707315859, −2.66572205261691285622720974500, −1.74827981053160497232937668648, −0.41078189771846528588792989008,
1.73209242325006347007664312388, 2.01912479504055884902578147107, 2.79468727966165255743692454918, 4.21774903482119067220351117500, 6.01352072582362418231128683313, 6.32147610151907112955129794951, 7.597879862549062760087605204013, 7.818015165129935626386779999947, 9.039433279833685621955422676842, 9.62745350269793614095689469430, 10.584457960541144371548544149475, 11.51801819404828420990435257518, 12.26352054501542777434143206564, 12.9083201599976211387646677448, 14.27779625502689334973755781966, 14.85105927518965421543706952914, 15.32522747119038339760657525429, 17.07936553384692470485683628365, 17.32736902548086831822555204995, 18.153676089954569059573101017212, 18.72791223892787792838013920473, 19.35490383732304853171586056808, 20.17053392379486256713224586887, 21.02398006601234558346448589913, 21.7433370491279344804571743562