L(s) = 1 | − 6.76i·2-s − 12.7i·3-s − 29.8·4-s + (−13.2 + 21.2i)5-s − 86.4·6-s + 5.97·7-s + 93.4i·8-s − 82.1·9-s + (143. + 89.3i)10-s + 18.7i·11-s + 380. i·12-s + 30.6i·13-s − 40.4i·14-s + (271. + 168. i)15-s + 155.·16-s − 280.·17-s + ⋯ |
L(s) = 1 | − 1.69i·2-s − 1.41i·3-s − 1.86·4-s + (−0.528 + 0.849i)5-s − 2.40·6-s + 0.121·7-s + 1.46i·8-s − 1.01·9-s + (1.43 + 0.893i)10-s + 0.155i·11-s + 2.64i·12-s + 0.181i·13-s − 0.206i·14-s + (1.20 + 0.749i)15-s + 0.608·16-s − 0.971·17-s + ⋯ |
Λ(s)=(=(115s/2ΓC(s)L(s)(0.833−0.552i)Λ(5−s)
Λ(s)=(=(115s/2ΓC(s+2)L(s)(0.833−0.552i)Λ(1−s)
Degree: |
2 |
Conductor: |
115
= 5⋅23
|
Sign: |
0.833−0.552i
|
Analytic conductor: |
11.8875 |
Root analytic conductor: |
3.44783 |
Motivic weight: |
4 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ115(114,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 115, ( :2), 0.833−0.552i)
|
Particular Values
L(25) |
≈ |
0.265094+0.0799513i |
L(21) |
≈ |
0.265094+0.0799513i |
L(3) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(13.2−21.2i)T |
| 23 | 1+(481.+219.i)T |
good | 2 | 1+6.76iT−16T2 |
| 3 | 1+12.7iT−81T2 |
| 7 | 1−5.97T+2.40e3T2 |
| 11 | 1−18.7iT−1.46e4T2 |
| 13 | 1−30.6iT−2.85e4T2 |
| 17 | 1+280.T+8.35e4T2 |
| 19 | 1−93.6iT−1.30e5T2 |
| 29 | 1−378.T+7.07e5T2 |
| 31 | 1−25.0T+9.23e5T2 |
| 37 | 1+2.50e3T+1.87e6T2 |
| 41 | 1−182.T+2.82e6T2 |
| 43 | 1−3.03e3T+3.41e6T2 |
| 47 | 1−2.99e3iT−4.87e6T2 |
| 53 | 1−3.12e3T+7.89e6T2 |
| 59 | 1+495.T+1.21e7T2 |
| 61 | 1+3.23e3iT−1.38e7T2 |
| 67 | 1+5.93e3T+2.01e7T2 |
| 71 | 1+8.19e3T+2.54e7T2 |
| 73 | 1+6.90e3iT−2.83e7T2 |
| 79 | 1−3.46e3iT−3.89e7T2 |
| 83 | 1+7.71e3T+4.74e7T2 |
| 89 | 1−2.86e3iT−6.27e7T2 |
| 97 | 1+1.52e3T+8.85e7T2 |
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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
−11.99597261157795625184799525260, −11.12193588216989091150538645877, −10.22776108075313501481106838785, −8.745663110581633329979635782230, −7.54168486046294545046189181485, −6.40642221657024911400631345893, −4.23149598696874856633382391412, −2.76015967441899488910601670980, −1.71974412646495322626749410634, −0.11974346179305530082886418936,
3.96095036413996811455380213674, 4.78907557384286616327298581306, 5.72756899710910252093142014399, 7.28324618371942487806551072564, 8.550604601254261833946188462114, 9.056648681887890549032867333872, 10.30527243236774313570614157502, 11.67499685370870310821050072302, 13.17569682843161815015939783839, 14.21320620194939977407638807099