L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 12-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + 21-s − 23-s + (−0.309 + 0.951i)24-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 12-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + 21-s − 23-s + (−0.309 + 0.951i)24-s + ⋯ |
Λ(s)=(=(1045s/2ΓR(s)L(s)(0.970−0.242i)Λ(1−s)
Λ(s)=(=(1045s/2ΓR(s)L(s)(0.970−0.242i)Λ(1−s)
Degree: |
1 |
Conductor: |
1045
= 5⋅11⋅19
|
Sign: |
0.970−0.242i
|
Analytic conductor: |
4.85295 |
Root analytic conductor: |
4.85295 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1045(569,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1045, (0: ), 0.970−0.242i)
|
Particular Values
L(21) |
≈ |
0.3429665136−0.04215966795i |
L(21) |
≈ |
0.3429665136−0.04215966795i |
L(1) |
≈ |
0.4548600805+0.2420944032i |
L(1) |
≈ |
0.4548600805+0.2420944032i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 11 | 1 |
| 19 | 1 |
good | 2 | 1+(−0.309+0.951i)T |
| 3 | 1+(−0.809+0.587i)T |
| 7 | 1+(−0.809−0.587i)T |
| 13 | 1+(−0.309+0.951i)T |
| 17 | 1+(0.309+0.951i)T |
| 23 | 1−T |
| 29 | 1+(−0.809−0.587i)T |
| 31 | 1+(−0.309+0.951i)T |
| 37 | 1+(−0.809−0.587i)T |
| 41 | 1+(−0.809+0.587i)T |
| 43 | 1+T |
| 47 | 1+(0.809−0.587i)T |
| 53 | 1+(0.309−0.951i)T |
| 59 | 1+(0.809+0.587i)T |
| 61 | 1+(−0.309−0.951i)T |
| 67 | 1+T |
| 71 | 1+(−0.309−0.951i)T |
| 73 | 1+(−0.809−0.587i)T |
| 79 | 1+(0.309−0.951i)T |
| 83 | 1+(0.309+0.951i)T |
| 89 | 1−T |
| 97 | 1+(0.309−0.951i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.81968324476723487957311540586, −20.60082141023963195426225471396, −20.01116218849914423279227453810, −18.987843003971627905963679338312, −18.635814574746116731558818700305, −17.86141851775092489574477261132, −17.12376749057658505741369682850, −16.32615380194318577350363266237, −15.511996133886287766074916217625, −14.14909128986421006927475523696, −13.30705915789712285109866401445, −12.61539973378782782820655544278, −12.082045691385163199127014084988, −11.34894368461731142042730790628, −10.38599893540885432645930863123, −9.78705421868169889618770593989, −8.82418021468702921860105418816, −7.78778916841854888412323373585, −7.068935968254565135860171008192, −5.773692298232011161499322739828, −5.25402688146794948079464901503, −3.99403219891259908495918356597, −2.88374406410455142618704620052, −2.12782144789846372403009793372, −0.85526618304564913751494492371,
0.24704737836385679077592904813, 1.66459471064338123885467229207, 3.66933362696989145048028768407, 4.158262511772334790760545061546, 5.23395302163544445132876545387, 6.06844597513559303626355842490, 6.71777203555932320399973222845, 7.48881663220829915117300817279, 8.69677310430059740064954050068, 9.53446169585495453490177437421, 10.15236996110709096596892693335, 10.83224141839716217578873938042, 12.02498679778995092295111560172, 12.84214694636383163963542677970, 13.83098501563859294849138918495, 14.61047527198421177118411483604, 15.48650431643691158748680704933, 16.21580708099084910906324452772, 16.73140338573600012563543865453, 17.32445852628715886251630743405, 18.15623208377893577686566774441, 19.08560711135058406810471938707, 19.71080943257989983986851891282, 20.86343393420598142587566690675, 21.87680307252536513399423759469