L(s) = 1 | + (−1.41 − 0.0832i)2-s + (−2.32 − 0.961i)3-s + (1.98 + 0.235i)4-s + (0.932 − 2.25i)5-s + (3.19 + 1.55i)6-s + 4.08i·7-s + (−2.78 − 0.497i)8-s + (2.34 + 2.34i)9-s + (−1.50 + 3.10i)10-s + (−5.01 − 2.07i)11-s + (−4.38 − 2.45i)12-s + (3.14 + 1.75i)13-s + (0.339 − 5.76i)14-s + (−4.33 + 4.33i)15-s + (3.88 + 0.934i)16-s − 5.02·17-s + ⋯ |
L(s) = 1 | + (−0.998 − 0.0588i)2-s + (−1.34 − 0.555i)3-s + (0.993 + 0.117i)4-s + (0.417 − 1.00i)5-s + (1.30 + 0.633i)6-s + 1.54i·7-s + (−0.984 − 0.175i)8-s + (0.781 + 0.781i)9-s + (−0.475 + 0.980i)10-s + (−1.51 − 0.626i)11-s + (−1.26 − 0.709i)12-s + (0.873 + 0.487i)13-s + (0.0908 − 1.53i)14-s + (−1.11 + 1.11i)15-s + (0.972 + 0.233i)16-s − 1.21·17-s + ⋯ |
Λ(s)=(=(416s/2ΓC(s)L(s)(0.648−0.760i)Λ(2−s)
Λ(s)=(=(416s/2ΓC(s+1/2)L(s)(0.648−0.760i)Λ(1−s)
Degree: |
2 |
Conductor: |
416
= 25⋅13
|
Sign: |
0.648−0.760i
|
Analytic conductor: |
3.32177 |
Root analytic conductor: |
1.82257 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ416(83,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 416, ( :1/2), 0.648−0.760i)
|
Particular Values
L(1) |
≈ |
0.367895+0.169752i |
L(21) |
≈ |
0.367895+0.169752i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.41+0.0832i)T |
| 13 | 1+(−3.14−1.75i)T |
good | 3 | 1+(2.32+0.961i)T+(2.12+2.12i)T2 |
| 5 | 1+(−0.932+2.25i)T+(−3.53−3.53i)T2 |
| 7 | 1−4.08iT−7T2 |
| 11 | 1+(5.01+2.07i)T+(7.77+7.77i)T2 |
| 17 | 1+5.02T+17T2 |
| 19 | 1+(0.314+0.759i)T+(−13.4+13.4i)T2 |
| 23 | 1+(−5.64−5.64i)T+23iT2 |
| 29 | 1+(1.61−3.89i)T+(−20.5−20.5i)T2 |
| 31 | 1+(−0.191−0.191i)T+31iT2 |
| 37 | 1+(−4.82−1.99i)T+(26.1+26.1i)T2 |
| 41 | 1−4.38T+41T2 |
| 43 | 1+(0.182+0.440i)T+(−30.4+30.4i)T2 |
| 47 | 1+(−6.13−6.13i)T+47iT2 |
| 53 | 1+(−3.34−8.06i)T+(−37.4+37.4i)T2 |
| 59 | 1+(0.901−2.17i)T+(−41.7−41.7i)T2 |
| 61 | 1+(−0.967+2.33i)T+(−43.1−43.1i)T2 |
| 67 | 1+(−1.21+0.503i)T+(47.3−47.3i)T2 |
| 71 | 1+1.22T+71T2 |
| 73 | 1+4.96iT−73T2 |
| 79 | 1−3.63T+79T2 |
| 83 | 1+(1.66+4.01i)T+(−58.6+58.6i)T2 |
| 89 | 1+5.69T+89T2 |
| 97 | 1+(6.00−6.00i)T−97iT2 |
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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
−11.19358897592591591987917242355, −10.77247501969833758091282689248, −9.092777153724654105984658612354, −8.961156980507726396799670378203, −7.75322008854259537956945968303, −6.48552843102054287656884714284, −5.69966280866006985640495185541, −5.15707925273424879032668574601, −2.60676737478445037974093800382, −1.23717439021483633095278275796,
0.46567380299628967764169754387, 2.59855126748119942754930928688, 4.27847190524267146618305842473, 5.56999320009487803520885237169, 6.59991580746071293194865023816, 7.13866454408819402721525954795, 8.253710210125213498670229491518, 9.796309165881864644850422045952, 10.45500037885933247296167498573, 10.82623568582672809717776569655