L(s) = 1 | + (1.67 + 0.967i)2-s + (−1.73 − 0.0524i)3-s + (0.871 + 1.50i)4-s + (2.26 + 1.30i)5-s + (−2.84 − 1.76i)6-s + 2.32i·7-s − 0.497i·8-s + (2.99 + 0.181i)9-s + (2.53 + 4.38i)10-s + (−4.66 − 2.69i)11-s + (−1.42 − 2.65i)12-s + (−1.31 − 3.35i)13-s + (−2.25 + 3.90i)14-s + (−3.85 − 2.38i)15-s + (2.22 − 3.85i)16-s + (0.835 − 1.44i)17-s + ⋯ |
L(s) = 1 | + (1.18 + 0.683i)2-s + (−0.999 − 0.0302i)3-s + (0.435 + 0.754i)4-s + (1.01 + 0.585i)5-s + (−1.16 − 0.719i)6-s + 0.880i·7-s − 0.175i·8-s + (0.998 + 0.0605i)9-s + (0.800 + 1.38i)10-s + (−1.40 − 0.812i)11-s + (−0.412 − 0.767i)12-s + (−0.366 − 0.930i)13-s + (−0.602 + 1.04i)14-s + (−0.995 − 0.615i)15-s + (0.556 − 0.963i)16-s + (0.202 − 0.351i)17-s + ⋯ |
Λ(s)=(=(117s/2ΓC(s)L(s)(0.540−0.841i)Λ(2−s)
Λ(s)=(=(117s/2ΓC(s+1/2)L(s)(0.540−0.841i)Λ(1−s)
Degree: |
2 |
Conductor: |
117
= 32⋅13
|
Sign: |
0.540−0.841i
|
Analytic conductor: |
0.934249 |
Root analytic conductor: |
0.966565 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ117(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 117, ( :1/2), 0.540−0.841i)
|
Particular Values
L(1) |
≈ |
1.32653+0.724240i |
L(21) |
≈ |
1.32653+0.724240i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.73+0.0524i)T |
| 13 | 1+(1.31+3.35i)T |
good | 2 | 1+(−1.67−0.967i)T+(1+1.73i)T2 |
| 5 | 1+(−2.26−1.30i)T+(2.5+4.33i)T2 |
| 7 | 1−2.32iT−7T2 |
| 11 | 1+(4.66+2.69i)T+(5.5+9.52i)T2 |
| 17 | 1+(−0.835+1.44i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−1.90−1.09i)T+(9.5+16.4i)T2 |
| 23 | 1+4.20T+23T2 |
| 29 | 1+(2.86−4.95i)T+(−14.5−25.1i)T2 |
| 31 | 1+(−7.55−4.35i)T+(15.5+26.8i)T2 |
| 37 | 1+(1.22−0.707i)T+(18.5−32.0i)T2 |
| 41 | 1−1.52iT−41T2 |
| 43 | 1−1.87T+43T2 |
| 47 | 1+(4.47−2.58i)T+(23.5−40.7i)T2 |
| 53 | 1+9.81T+53T2 |
| 59 | 1+(−6.59+3.80i)T+(29.5−51.0i)T2 |
| 61 | 1+0.934T+61T2 |
| 67 | 1−2.06iT−67T2 |
| 71 | 1+(10.9+6.33i)T+(35.5+61.4i)T2 |
| 73 | 1−12.6iT−73T2 |
| 79 | 1+(−3.46−6.00i)T+(−39.5+68.4i)T2 |
| 83 | 1+(−7.95+4.59i)T+(41.5−71.8i)T2 |
| 89 | 1+(−8.62+4.97i)T+(44.5−77.0i)T2 |
| 97 | 1−2.35iT−97T2 |
show more | |
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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
−13.65111305469742957289923205089, −12.87723232804858255165669552295, −11.99896632143911474525815798617, −10.59690611472968018797929807808, −9.838742422422758899305743672476, −7.82770182546127502434865868231, −6.45373917833328885835314751871, −5.62889259985791955616401498338, −5.13303023783236174181216528882, −2.93383560022990523218683437339,
2.01543149161924208781164402229, 4.27608720021242016642176760662, 5.07800700022999542993751229556, 6.10915612479576197141167751801, 7.64686198680437349400644367036, 9.754567278707335321998339446735, 10.40669720665300417183274236502, 11.57010553904228217936695315096, 12.46508478279473700112932819792, 13.31822568946150212657307382224