L(s) = 1 | + (−0.615 + 1.06i)2-s + (1.01 − 1.75i)3-s + (0.241 + 0.417i)4-s + (−1.31 + 2.27i)5-s + (1.24 + 2.16i)6-s + 3.75·7-s − 3.05·8-s + (−0.555 − 0.962i)9-s + (−1.62 − 2.80i)10-s − 11-s + 0.978·12-s + (0.824 + 1.42i)13-s + (−2.31 + 4.00i)14-s + (2.66 + 4.62i)15-s + (1.40 − 2.42i)16-s + (0.706 − 1.22i)17-s + ⋯ |
L(s) = 1 | + (−0.435 + 0.754i)2-s + (0.585 − 1.01i)3-s + (0.120 + 0.208i)4-s + (−0.588 + 1.01i)5-s + (0.509 + 0.883i)6-s + 1.42·7-s − 1.08·8-s + (−0.185 − 0.320i)9-s + (−0.512 − 0.887i)10-s − 0.301·11-s + 0.282·12-s + (0.228 + 0.396i)13-s + (−0.618 + 1.07i)14-s + (0.688 + 1.19i)15-s + (0.350 − 0.606i)16-s + (0.171 − 0.296i)17-s + ⋯ |
Λ(s)=(=(209s/2ΓC(s)L(s)(0.466−0.884i)Λ(2−s)
Λ(s)=(=(209s/2ΓC(s+1/2)L(s)(0.466−0.884i)Λ(1−s)
Degree: |
2 |
Conductor: |
209
= 11⋅19
|
Sign: |
0.466−0.884i
|
Analytic conductor: |
1.66887 |
Root analytic conductor: |
1.29184 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ209(144,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 209, ( :1/2), 0.466−0.884i)
|
Particular Values
L(1) |
≈ |
1.04314+0.629207i |
L(21) |
≈ |
1.04314+0.629207i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1+T |
| 19 | 1+(−4.29−0.756i)T |
good | 2 | 1+(0.615−1.06i)T+(−1−1.73i)T2 |
| 3 | 1+(−1.01+1.75i)T+(−1.5−2.59i)T2 |
| 5 | 1+(1.31−2.27i)T+(−2.5−4.33i)T2 |
| 7 | 1−3.75T+7T2 |
| 13 | 1+(−0.824−1.42i)T+(−6.5+11.2i)T2 |
| 17 | 1+(−0.706+1.22i)T+(−8.5−14.7i)T2 |
| 23 | 1+(−1.21−2.10i)T+(−11.5+19.9i)T2 |
| 29 | 1+(3.60+6.24i)T+(−14.5+25.1i)T2 |
| 31 | 1+9.66T+31T2 |
| 37 | 1+9.68T+37T2 |
| 41 | 1+(−6.11+10.5i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−2.96+5.14i)T+(−21.5−37.2i)T2 |
| 47 | 1+(0.0433+0.0750i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−0.599−1.03i)T+(−26.5+45.8i)T2 |
| 59 | 1+(−1.36+2.35i)T+(−29.5−51.0i)T2 |
| 61 | 1+(5.87+10.1i)T+(−30.5+52.8i)T2 |
| 67 | 1+(2.07+3.59i)T+(−33.5+58.0i)T2 |
| 71 | 1+(1.31−2.28i)T+(−35.5−61.4i)T2 |
| 73 | 1+(6.11−10.5i)T+(−36.5−63.2i)T2 |
| 79 | 1+(1.58−2.75i)T+(−39.5−68.4i)T2 |
| 83 | 1+1.23T+83T2 |
| 89 | 1+(−7.60−13.1i)T+(−44.5+77.0i)T2 |
| 97 | 1+(−1.66+2.87i)T+(−48.5−84.0i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−12.45955051931341631924776393458, −11.57975278081843056288355504824, −10.85376931083171953204749780588, −9.107099427090597252845020317792, −8.112917393058000103696042170415, −7.37834369787984073207245460700, −7.15360851495513252783197712459, −5.53224914056613876681536081486, −3.53386260860338388534078763862, −2.10016784650197640496502675543,
1.38658539029827566677919895529, 3.25099244481248862893182803798, 4.57475651232126319194076607416, 5.45624772389531176816688122049, 7.62081768612596837170486411727, 8.702991232161491008548080678627, 9.149980527245543426968420478402, 10.37196574718643269460931658264, 11.04971354888269063655894759781, 11.94072977010332620142985934068