↑

Properties

Label 2-845-65.28-c1-0-42
Degree 22
Conductor 845845
Sign 0.824+0.566i0.824 + 0.566i
Analytic cond. 6.747356.74735
Root an. cond. 2.597562.59756
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 + 1.04i)2-s + (0.713 − 2.66i)3-s + (1.17 − 2.04i)4-s + (2.22 + 0.194i)5-s + (1.48 + 5.55i)6-s + (−1.45 + 2.52i)7-s + 0.750i·8-s + (−3.97 − 2.29i)9-s + (−4.23 + 1.97i)10-s + (0.00681 − 0.0254i)11-s + (−4.59 − 4.59i)12-s − 6.07i·14-s + (2.10 − 5.79i)15-s + (1.57 + 2.72i)16-s + (2.76 − 0.741i)17-s + 9.58·18-s + ⋯
L(s)  = 1  + (−1.27 + 0.738i)2-s + (0.411 − 1.53i)3-s + (0.589 − 1.02i)4-s + (0.996 + 0.0869i)5-s + (0.607 + 2.26i)6-s + (−0.550 + 0.952i)7-s + 0.265i·8-s + (−1.32 − 0.765i)9-s + (−1.33 + 0.624i)10-s + (0.00205 − 0.00767i)11-s + (−1.32 − 1.32i)12-s − 1.62i·14-s + (0.543 − 1.49i)15-s + (0.393 + 0.682i)16-s + (0.671 − 0.179i)17-s + 2.26·18-s + ⋯

Functional equation

Λ(s)=(845s/2 ΓC(s) L(s)=((0.824+0.566i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(845s/2 ΓC(s+1/2) L(s)=((0.824+0.566i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 845845    =    5⋅1325 \cdot 13^{2}
Sign: 0.824+0.566i0.824 + 0.566i
Analytic conductor: 6.747356.74735
Root analytic conductor: 2.597562.59756
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ845(418,⋅)\chi_{845} (418, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 845, ( :1/2), 0.824+0.566i)(2,\ 845,\ (\ :1/2),\ 0.824 + 0.566i)

Particular Values

L(1)L(1) ≈\approx 1.00299−0.311358i1.00299 - 0.311358i
L(12)L(\frac12) ≈\approx 1.00299−0.311358i1.00299 - 0.311358i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(−2.22−0.194i)T 1 + (-2.22 - 0.194i)T
13 1 1
good2 1+(1.80−1.04i)T+(1−1.73i)T2 1 + (1.80 - 1.04i)T + (1 - 1.73i)T^{2}
3 1+(−0.713+2.66i)T+(−2.59−1.5i)T2 1 + (-0.713 + 2.66i)T + (-2.59 - 1.5i)T^{2}
7 1+(1.45−2.52i)T+(−3.5−6.06i)T2 1 + (1.45 - 2.52i)T + (-3.5 - 6.06i)T^{2}
11 1+(−0.00681+0.0254i)T+(−9.52−5.5i)T2 1 + (-0.00681 + 0.0254i)T + (-9.52 - 5.5i)T^{2}
17 1+(−2.76+0.741i)T+(14.7−8.5i)T2 1 + (-2.76 + 0.741i)T + (14.7 - 8.5i)T^{2}
19 1+(−4.62+1.23i)T+(16.4−9.5i)T2 1 + (-4.62 + 1.23i)T + (16.4 - 9.5i)T^{2}
23 1+(−0.358−0.0961i)T+(19.9+11.5i)T2 1 + (-0.358 - 0.0961i)T + (19.9 + 11.5i)T^{2}
29 1+(−3.62+2.09i)T+(14.5−25.1i)T2 1 + (-3.62 + 2.09i)T + (14.5 - 25.1i)T^{2}
31 1+(0.835−0.835i)T−31iT2 1 + (0.835 - 0.835i)T - 31iT^{2}
37 1+(3.22+5.58i)T+(−18.5+32.0i)T2 1 + (3.22 + 5.58i)T + (-18.5 + 32.0i)T^{2}
41 1+(−7.57−2.02i)T+(35.5+20.5i)T2 1 + (-7.57 - 2.02i)T + (35.5 + 20.5i)T^{2}
43 1+(1.79+6.69i)T+(−37.2+21.5i)T2 1 + (1.79 + 6.69i)T + (-37.2 + 21.5i)T^{2}
47 1−0.833T+47T2 1 - 0.833T + 47T^{2}
53 1+(−0.902−0.902i)T+53iT2 1 + (-0.902 - 0.902i)T + 53iT^{2}
59 1+(−0.387−1.44i)T+(−51.0+29.5i)T2 1 + (-0.387 - 1.44i)T + (-51.0 + 29.5i)T^{2}
61 1+(−5.35+9.26i)T+(−30.5−52.8i)T2 1 + (-5.35 + 9.26i)T + (-30.5 - 52.8i)T^{2}
67 1+(10.6−6.15i)T+(33.5−58.0i)T2 1 + (10.6 - 6.15i)T + (33.5 - 58.0i)T^{2}
71 1+(−0.957−3.57i)T+(−61.4+35.5i)T2 1 + (-0.957 - 3.57i)T + (-61.4 + 35.5i)T^{2}
73 1+15.0iT−73T2 1 + 15.0iT - 73T^{2}
79 1−4.25iT−79T2 1 - 4.25iT - 79T^{2}
83 1−1.31T+83T2 1 - 1.31T + 83T^{2}
89 1+(−3.23−0.867i)T+(77.0+44.5i)T2 1 + (-3.23 - 0.867i)T + (77.0 + 44.5i)T^{2}
97 1+(0.351+0.202i)T+(48.5+84.0i)T2 1 + (0.351 + 0.202i)T + (48.5 + 84.0i)T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.615360781750115715313516567327, −9.139865286826044365217477761932, −8.403155880299672131094053716921, −7.52701919399634140710206293215, −6.90937481199772156614499156183, −6.12413685945732025729100571811, −5.51280118045572628723761354095, −3.06161995190510623134375368638, −2.06308372099365844265072094730, −0.911880240108704267850344251340, 1.18964357802513648038031865662, 2.76682771391222428133153897793, 3.53116181060236376111142002732, 4.77390445829057680314560997560, 5.74868782352993138855240041769, 7.13908397965510069324652780409, 8.210230092618324643780546905821, 9.053894157047679628254405158517, 9.664267213058741828235188629392, 10.13357366197801514805161815910

Graph of the ZZ-function along the critical line