L(s) = 1 | + (−1.35 − 0.410i)2-s + (−1.57 − 1.96i)3-s + (1.66 + 1.11i)4-s + (1.55 − 1.23i)5-s + (1.31 + 3.31i)6-s + (2.03 − 1.69i)7-s + (−1.79 − 2.18i)8-s + (−0.745 + 3.26i)9-s + (−2.61 + 1.03i)10-s + (3.09 − 0.707i)11-s + (−0.423 − 5.02i)12-s + (−5.38 + 1.22i)13-s + (−3.44 + 1.45i)14-s + (−4.88 − 1.11i)15-s + (1.53 + 3.69i)16-s + (2.48 − 5.15i)17-s + ⋯ |
L(s) = 1 | + (−0.956 − 0.290i)2-s + (−0.906 − 1.13i)3-s + (0.831 + 0.555i)4-s + (0.695 − 0.554i)5-s + (0.537 + 1.35i)6-s + (0.768 − 0.639i)7-s + (−0.634 − 0.772i)8-s + (−0.248 + 1.08i)9-s + (−0.826 + 0.328i)10-s + (0.934 − 0.213i)11-s + (−0.122 − 1.44i)12-s + (−1.49 + 0.340i)13-s + (−0.921 + 0.388i)14-s + (−1.26 − 0.287i)15-s + (0.382 + 0.923i)16-s + (0.601 − 1.24i)17-s + ⋯ |
Λ(s)=(=(196s/2ΓC(s)L(s)(−0.720+0.693i)Λ(2−s)
Λ(s)=(=(196s/2ΓC(s+1/2)L(s)(−0.720+0.693i)Λ(1−s)
Degree: |
2 |
Conductor: |
196
= 22⋅72
|
Sign: |
−0.720+0.693i
|
Analytic conductor: |
1.56506 |
Root analytic conductor: |
1.25102 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ196(111,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 196, ( :1/2), −0.720+0.693i)
|
Particular Values
L(1) |
≈ |
0.241972−0.600233i |
L(21) |
≈ |
0.241972−0.600233i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.35+0.410i)T |
| 7 | 1+(−2.03+1.69i)T |
good | 3 | 1+(1.57+1.96i)T+(−0.667+2.92i)T2 |
| 5 | 1+(−1.55+1.23i)T+(1.11−4.87i)T2 |
| 11 | 1+(−3.09+0.707i)T+(9.91−4.77i)T2 |
| 13 | 1+(5.38−1.22i)T+(11.7−5.64i)T2 |
| 17 | 1+(−2.48+5.15i)T+(−10.5−13.2i)T2 |
| 19 | 1+7.59T+19T2 |
| 23 | 1+(0.751+1.55i)T+(−14.3+17.9i)T2 |
| 29 | 1+(−0.626−0.301i)T+(18.0+22.6i)T2 |
| 31 | 1−8.05T+31T2 |
| 37 | 1+(−4.24−2.04i)T+(23.0+28.9i)T2 |
| 41 | 1+(0.0970−0.0773i)T+(9.12−39.9i)T2 |
| 43 | 1+(−3.53−2.81i)T+(9.56+41.9i)T2 |
| 47 | 1+(0.299+1.31i)T+(−42.3+20.3i)T2 |
| 53 | 1+(−0.183+0.0884i)T+(33.0−41.4i)T2 |
| 59 | 1+(1.40−1.76i)T+(−13.1−57.5i)T2 |
| 61 | 1+(−0.524+1.08i)T+(−38.0−47.6i)T2 |
| 67 | 1+3.71iT−67T2 |
| 71 | 1+(3.42+7.11i)T+(−44.2+55.5i)T2 |
| 73 | 1+(−4.61−1.05i)T+(65.7+31.6i)T2 |
| 79 | 1−9.99iT−79T2 |
| 83 | 1+(−0.712+3.12i)T+(−74.7−36.0i)T2 |
| 89 | 1+(−14.8−3.38i)T+(80.1+38.6i)T2 |
| 97 | 1+3.30iT−97T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.97585010062639679658597880447, −11.34338368798777330622395574069, −10.17424725409516045080507536133, −9.203481197465715730878133652958, −7.959737762252955192199130765441, −7.04860162187499576956628016672, −6.21964966996212499539649870352, −4.71066106995781362659516513879, −2.12514264263622905200503138540, −0.882813444233620004289886396439,
2.21229052647269112522790722709, 4.51207884083504168146727279472, 5.74015221537554484640701029819, 6.43826795972425448937625474607, 7.978070599038829003988284462049, 9.155360396021029820823082270524, 10.13358827666362227220979468298, 10.50031911030801684264612671662, 11.57773595532288350880826193071, 12.34825800178513904051970835048