L(s) = 1 | − 3·3-s + 3i·5-s − i·7-s + 6·9-s + 4i·11-s + (−2 − 3i)13-s − 9i·15-s − 5·17-s − 6i·19-s + 3i·21-s − 6·23-s − 4·25-s − 9·27-s − 4·29-s − 12i·33-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1.34i·5-s − 0.377i·7-s + 2·9-s + 1.20i·11-s + (−0.554 − 0.832i)13-s − 2.32i·15-s − 1.21·17-s − 1.37i·19-s + 0.654i·21-s − 1.25·23-s − 0.800·25-s − 1.73·27-s − 0.742·29-s − 2.08i·33-s + ⋯ |
Λ(s)=(=(416s/2ΓC(s)L(s)(−0.832+0.554i)Λ(2−s)
Λ(s)=(=(416s/2ΓC(s+1/2)L(s)(−0.832+0.554i)Λ(1−s)
Degree: |
2 |
Conductor: |
416
= 25⋅13
|
Sign: |
−0.832+0.554i
|
Analytic conductor: |
3.32177 |
Root analytic conductor: |
1.82257 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ416(129,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
1
|
Selberg data: |
(2, 416, ( :1/2), −0.832+0.554i)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 13 | 1+(2+3i)T |
good | 3 | 1+3T+3T2 |
| 5 | 1−3iT−5T2 |
| 7 | 1+iT−7T2 |
| 11 | 1−4iT−11T2 |
| 17 | 1+5T+17T2 |
| 19 | 1+6iT−19T2 |
| 23 | 1+6T+23T2 |
| 29 | 1+4T+29T2 |
| 31 | 1−31T2 |
| 37 | 1+3iT−37T2 |
| 41 | 1+12iT−41T2 |
| 43 | 1+3T+43T2 |
| 47 | 1−7iT−47T2 |
| 53 | 1+2T+53T2 |
| 59 | 1+2iT−59T2 |
| 61 | 1+12T+61T2 |
| 67 | 1+4iT−67T2 |
| 71 | 1−11iT−71T2 |
| 73 | 1−6iT−73T2 |
| 79 | 1−6T+79T2 |
| 83 | 1+10iT−83T2 |
| 89 | 1−6iT−89T2 |
| 97 | 1−97T2 |
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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
−10.72301841796029202465661168657, −10.45065652592070170228992426050, −9.428483445700316367073144700927, −7.46646046416840431551234300796, −7.00599394138295271063141268226, −6.20214531995203480907610222723, −5.10019492592589482081221636138, −4.12187527325456957596345917235, −2.31848070166838050919277286340, 0,
1.58958060048972467946282313599, 4.11737137647606119361774060872, 4.96700719456017829763534961263, 5.83759866425758622327451266892, 6.47010685473241215068343474096, 7.944595749674271891760273873132, 8.924066083756116351624652023821, 9.867705803055531930194665540861, 10.90058415489492309293663788560