L(s) = 1 | + (0.709 − 0.704i)2-s + (−0.949 + 0.313i)3-s + (0.00797 − 0.999i)4-s + (0.140 − 0.990i)5-s + (−0.453 + 0.891i)6-s + (0.631 − 0.775i)7-s + (−0.698 − 0.715i)8-s + (0.803 − 0.595i)9-s + (−0.597 − 0.801i)10-s + (0.768 − 0.639i)11-s + (0.305 + 0.952i)12-s + (−0.875 + 0.483i)13-s + (−0.0981 − 0.995i)14-s + (0.177 + 0.984i)15-s + (−0.999 − 0.0159i)16-s + (0.242 − 0.970i)17-s + ⋯ |
L(s) = 1 | + (0.709 − 0.704i)2-s + (−0.949 + 0.313i)3-s + (0.00797 − 0.999i)4-s + (0.140 − 0.990i)5-s + (−0.453 + 0.891i)6-s + (0.631 − 0.775i)7-s + (−0.698 − 0.715i)8-s + (0.803 − 0.595i)9-s + (−0.597 − 0.801i)10-s + (0.768 − 0.639i)11-s + (0.305 + 0.952i)12-s + (−0.875 + 0.483i)13-s + (−0.0981 − 0.995i)14-s + (0.177 + 0.984i)15-s + (−0.999 − 0.0159i)16-s + (0.242 − 0.970i)17-s + ⋯ |
Λ(s)=(=(4729s/2ΓR(s)L(s)(−0.540+0.841i)Λ(1−s)
Λ(s)=(=(4729s/2ΓR(s)L(s)(−0.540+0.841i)Λ(1−s)
Degree: |
1 |
Conductor: |
4729
|
Sign: |
−0.540+0.841i
|
Analytic conductor: |
21.9613 |
Root analytic conductor: |
21.9613 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ4729(3282,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 4729, (0: ), −0.540+0.841i)
|
Particular Values
L(21) |
≈ |
−0.6354782078−1.164000206i |
L(21) |
≈ |
−0.6354782078−1.164000206i |
L(1) |
≈ |
0.7623311360−0.8398107802i |
L(1) |
≈ |
0.7623311360−0.8398107802i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 4729 | 1 |
good | 2 | 1+(0.709−0.704i)T |
| 3 | 1+(−0.949+0.313i)T |
| 5 | 1+(0.140−0.990i)T |
| 7 | 1+(0.631−0.775i)T |
| 11 | 1+(0.768−0.639i)T |
| 13 | 1+(−0.875+0.483i)T |
| 17 | 1+(0.242−0.970i)T |
| 19 | 1+(0.749+0.661i)T |
| 23 | 1+(−0.0637−0.997i)T |
| 29 | 1+(−0.704+0.709i)T |
| 31 | 1+(0.956+0.290i)T |
| 37 | 1+(−0.129−0.991i)T |
| 41 | 1+(0.174+0.984i)T |
| 43 | 1+(−0.972+0.234i)T |
| 47 | 1+(0.881+0.472i)T |
| 53 | 1+(−0.450−0.892i)T |
| 59 | 1+(−0.713−0.700i)T |
| 61 | 1+(−0.685−0.728i)T |
| 67 | 1+(−0.647+0.762i)T |
| 71 | 1+(0.0132+0.999i)T |
| 73 | 1+(−0.370+0.928i)T |
| 79 | 1+(0.866−0.5i)T |
| 83 | 1+(−0.375+0.926i)T |
| 89 | 1+(−0.987−0.156i)T |
| 97 | 1+(−0.631+0.775i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−18.28144381563708663102662888552, −17.753355904282104530428475310750, −17.187917092061963060854951787439, −16.903148543532777308805201611928, −15.49408684159581564896186522213, −15.319220709939558329674405484594, −14.82430136133392842399370258876, −13.830944313505608489922842111951, −13.42318464330508339663797481213, −12.294450083267708204520924416114, −12.00345758887371008104182378565, −11.48532606241591183272053481821, −10.64496978224513914421600314392, −9.8188744871852133399053009238, −9.0136970104477367095319385073, −7.71439972194042586168626537899, −7.60756841734879893303719492041, −6.73259795846136359445604793567, −6.07209084802445589160245270754, −5.550482528549180787045186714121, −4.84147057016961026874923498617, −4.10679121625674401296785237563, −3.11996258013803350331912114771, −2.27668833180681335880955951524, −1.52518846005755041506162427586,
0.31721934354165124204778947743, 1.17420349780905446851331756926, 1.620833931418219983385442478164, 2.92976835321432201679966906921, 3.96050601543604621438799261659, 4.393679853739126585436584474596, 5.08703818385370654333498610883, 5.51039552766609448883297201904, 6.46679449720705400379981889214, 7.10484433937828798782416713055, 8.14750720691657749986300646466, 9.25147443943877342117121439361, 9.652017211409536773095795923939, 10.36641547111430889299146858860, 11.201616409473186538982022183, 11.65323739736837849550791333682, 12.23462339820773742519988607005, 12.77126240193625273017533999045, 13.73426670892054171386230529093, 14.209252936323112123254514474784, 14.80177748312658213539164527804, 15.94243116771142902650669064465, 16.42377058231898290425587615795, 16.89730032468917199354199385877, 17.67410988547640522654713066309