L(s) = 1 | + (0.124 + 0.992i)2-s + (0.733 + 0.680i)3-s + (−0.969 + 0.246i)4-s + (0.411 + 0.911i)5-s + (−0.583 + 0.811i)6-s + (0.542 − 0.840i)7-s + (−0.365 − 0.930i)8-s + (0.0747 + 0.997i)9-s + (−0.853 + 0.521i)10-s + (−0.911 + 0.411i)11-s + (−0.878 − 0.478i)12-s + (−0.866 + 0.5i)13-s + (0.900 + 0.433i)14-s + (−0.318 + 0.947i)15-s + (0.878 − 0.478i)16-s + (−0.478 − 0.878i)17-s + ⋯ |
L(s) = 1 | + (0.124 + 0.992i)2-s + (0.733 + 0.680i)3-s + (−0.969 + 0.246i)4-s + (0.411 + 0.911i)5-s + (−0.583 + 0.811i)6-s + (0.542 − 0.840i)7-s + (−0.365 − 0.930i)8-s + (0.0747 + 0.997i)9-s + (−0.853 + 0.521i)10-s + (−0.911 + 0.411i)11-s + (−0.878 − 0.478i)12-s + (−0.866 + 0.5i)13-s + (0.900 + 0.433i)14-s + (−0.318 + 0.947i)15-s + (0.878 − 0.478i)16-s + (−0.478 − 0.878i)17-s + ⋯ |
Λ(s)=(=(1009s/2ΓR(s)L(s)(−0.791−0.610i)Λ(1−s)
Λ(s)=(=(1009s/2ΓR(s)L(s)(−0.791−0.610i)Λ(1−s)
Degree: |
1 |
Conductor: |
1009
|
Sign: |
−0.791−0.610i
|
Analytic conductor: |
4.68577 |
Root analytic conductor: |
4.68577 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1009(101,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 1009, (0: ), −0.791−0.610i)
|
Particular Values
L(21) |
≈ |
−0.4592565623+1.346937111i |
L(21) |
≈ |
−0.4592565623+1.346937111i |
L(1) |
≈ |
0.6579525861+1.004145273i |
L(1) |
≈ |
0.6579525861+1.004145273i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 1009 | 1 |
good | 2 | 1+(0.124+0.992i)T |
| 3 | 1+(0.733+0.680i)T |
| 5 | 1+(0.411+0.911i)T |
| 7 | 1+(0.542−0.840i)T |
| 11 | 1+(−0.911+0.411i)T |
| 13 | 1+(−0.866+0.5i)T |
| 17 | 1+(−0.478−0.878i)T |
| 19 | 1+(0.866+0.5i)T |
| 23 | 1+(0.149+0.988i)T |
| 29 | 1+(−0.456+0.889i)T |
| 31 | 1+(−0.992−0.124i)T |
| 37 | 1+(0.939+0.342i)T |
| 41 | 1+(−0.939−0.342i)T |
| 43 | 1+(−0.866+0.5i)T |
| 47 | 1+(−0.294−0.955i)T |
| 53 | 1+(0.603+0.797i)T |
| 59 | 1+(−0.433−0.900i)T |
| 61 | 1+(−0.680−0.733i)T |
| 67 | 1+(0.542+0.840i)T |
| 71 | 1+(0.124+0.992i)T |
| 73 | 1+(−0.563+0.826i)T |
| 79 | 1+(0.947+0.318i)T |
| 83 | 1+(0.521−0.853i)T |
| 89 | 1+(0.811+0.583i)T |
| 97 | 1+(0.715+0.698i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−21.08371614621401008896364390600, −20.30817642993908111491920276072, −19.807333249695962147079581743893, −18.90949647223242553294743684579, −18.11441366542944200016643540478, −17.73514533263546463856212034890, −16.64389413889616910448656322994, −15.24388713136090212179487980493, −14.76045426227905854590108614927, −13.64507742521945761707775596803, −13.14152334737967011410714243596, −12.46981080611935852504070578883, −11.86092733242524632131262141422, −10.78497495343239361233371871751, −9.726194942853991769386765067341, −9.036475752487302519061316783539, −8.33059207696030290650893326799, −7.73513137583817518833800825375, −6.07465254832670418606744804000, −5.28428835102419505728507337697, −4.50358284808896982488116149000, −3.17337309738644207280616185816, −2.34553342338384439220382849341, −1.74797065676135816017784595100, −0.49226702022503345821614549239,
1.84937398104984311600250929908, 3.02735058084224014542440513099, 3.83048208831743930330563146021, 4.930396806708529792226501350875, 5.36293266112586503875272191408, 6.94913140535766580837439577656, 7.377766157912451291300016146424, 8.04373148218290487409146808633, 9.34355401375698315256639908851, 9.80062256090819925556648111878, 10.60265583185428644625938980354, 11.609567407171027786490028832117, 13.165185808864485226579098856839, 13.6733802557288772785677564009, 14.41572469555113284628260184385, 14.87461575902392429717913298110, 15.68601995687094369625140843215, 16.519143734619400575935413652462, 17.255108993893578778045862771001, 18.2173792057222369074268760197, 18.650248795675073748432092757052, 19.93340810779610483489131438821, 20.54921231125010337941899387231, 21.72608091014272193274275701465, 21.883944535439646439726094828784