L(s) = 1 | + (4 + 6.92i)2-s + (−15.9 − 27.6i)3-s + (−31.9 + 55.4i)4-s − 54.4·5-s + (127. − 220. i)6-s + (556. − 963. i)7-s − 511.·8-s + (585. − 1.01e3i)9-s + (−217. − 377. i)10-s + (−3.56e3 − 6.17e3i)11-s + 2.03e3·12-s + (7.56e3 + 2.34e3i)13-s + 8.90e3·14-s + (867. + 1.50e3i)15-s + (−2.04e3 − 3.54e3i)16-s + (1.02e4 − 1.77e4i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.340 − 0.590i)3-s + (−0.249 + 0.433i)4-s − 0.194·5-s + (0.240 − 0.417i)6-s + (0.613 − 1.06i)7-s − 0.353·8-s + (0.267 − 0.463i)9-s + (−0.0688 − 0.119i)10-s + (−0.807 − 1.39i)11-s + 0.340·12-s + (0.955 + 0.296i)13-s + 0.867·14-s + (0.0663 + 0.114i)15-s + (−0.125 − 0.216i)16-s + (0.506 − 0.876i)17-s + ⋯ |
Λ(s)=(=(26s/2ΓC(s)L(s)(0.471+0.881i)Λ(8−s)
Λ(s)=(=(26s/2ΓC(s+7/2)L(s)(0.471+0.881i)Λ(1−s)
Degree: |
2 |
Conductor: |
26
= 2⋅13
|
Sign: |
0.471+0.881i
|
Analytic conductor: |
8.12201 |
Root analytic conductor: |
2.84991 |
Motivic weight: |
7 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ26(3,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 26, ( :7/2), 0.471+0.881i)
|
Particular Values
L(4) |
≈ |
1.32876−0.796255i |
L(21) |
≈ |
1.32876−0.796255i |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−4−6.92i)T |
| 13 | 1+(−7.56e3−2.34e3i)T |
good | 3 | 1+(15.9+27.6i)T+(−1.09e3+1.89e3i)T2 |
| 5 | 1+54.4T+7.81e4T2 |
| 7 | 1+(−556.+963.i)T+(−4.11e5−7.13e5i)T2 |
| 11 | 1+(3.56e3+6.17e3i)T+(−9.74e6+1.68e7i)T2 |
| 17 | 1+(−1.02e4+1.77e4i)T+(−2.05e8−3.55e8i)T2 |
| 19 | 1+(1.40e4−2.43e4i)T+(−4.46e8−7.74e8i)T2 |
| 23 | 1+(1.69e4+2.93e4i)T+(−1.70e9+2.94e9i)T2 |
| 29 | 1+(−8.77e4−1.51e5i)T+(−8.62e9+1.49e10i)T2 |
| 31 | 1−1.56e4T+2.75e10T2 |
| 37 | 1+(998.+1.72e3i)T+(−4.74e10+8.22e10i)T2 |
| 41 | 1+(1.11e5+1.93e5i)T+(−9.73e10+1.68e11i)T2 |
| 43 | 1+(−2.68e5+4.65e5i)T+(−1.35e11−2.35e11i)T2 |
| 47 | 1−5.42e5T+5.06e11T2 |
| 53 | 1−1.85e6T+1.17e12T2 |
| 59 | 1+(−6.65e5+1.15e6i)T+(−1.24e12−2.15e12i)T2 |
| 61 | 1+(1.43e6−2.48e6i)T+(−1.57e12−2.72e12i)T2 |
| 67 | 1+(−1.41e6−2.45e6i)T+(−3.03e12+5.24e12i)T2 |
| 71 | 1+(−8.00e5+1.38e6i)T+(−4.54e12−7.87e12i)T2 |
| 73 | 1−1.39e6T+1.10e13T2 |
| 79 | 1+2.33e6T+1.92e13T2 |
| 83 | 1−2.37e6T+2.71e13T2 |
| 89 | 1+(3.52e6+6.10e6i)T+(−2.21e13+3.83e13i)T2 |
| 97 | 1+(4.25e6−7.37e6i)T+(−4.03e13−6.99e13i)T2 |
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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
−15.84032261032378979759255502373, −14.17034724646112985847384806431, −13.41791173636955681392993540769, −11.96292496787501535742903930285, −10.63579854752849494014744812124, −8.412601132366499250411333218446, −7.20767196296221803248795093025, −5.78443546173782474486460611623, −3.83752993980241522668129601175, −0.77624088404093377737204112699,
2.10824556057425073623084577750, 4.36270168228763664302675118250, 5.61392653010655724868642819762, 8.082845610661017241180821525442, 9.832907861596866597401421165562, 10.94936682865697185767004653392, 12.14591695924005880568788672485, 13.35578815406339104687970231013, 15.17594893357944776738699853944, 15.61343049491279625869924254534