↑

Properties

Label 2-936-104.101-c1-0-7
Degree 22
Conductor 936936
Sign 0.999−0.00641i0.999 - 0.00641i
Analytic cond. 7.473997.47399
Root an. cond. 2.733862.73386
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  + (−0.366 − 1.36i)2-s + (−1.73 + i)4-s − 3.73·5-s + (−3 − 1.73i)7-s + (2 + 1.99i)8-s + (1.36 + 5.09i)10-s + (−1 − 1.73i)11-s + (−2.59 − 2.5i)13-s + (−1.26 + 4.73i)14-s + (1.99 − 3.46i)16-s + (−0.232 + 0.401i)17-s + (0.633 − 1.09i)19-s + (6.46 − 3.73i)20-s + (−1.99 + 2i)22-s + (4.09 + 7.09i)23-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.5i)4-s − 1.66·5-s + (−1.13 − 0.654i)7-s + (0.707 + 0.707i)8-s + (0.431 + 1.61i)10-s + (−0.301 − 0.522i)11-s + (−0.720 − 0.693i)13-s + (−0.338 + 1.26i)14-s + (0.499 − 0.866i)16-s + (−0.0562 + 0.0974i)17-s + (0.145 − 0.251i)19-s + (1.44 − 0.834i)20-s + (−0.426 + 0.426i)22-s + (0.854 + 1.48i)23-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.999−0.00641i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00641i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((0.999−0.00641i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00641i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.999−0.00641i0.999 - 0.00641i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(829,⋅)\chi_{936} (829, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 0.999−0.00641i)(2,\ 936,\ (\ :1/2),\ 0.999 - 0.00641i)

Particular Values

L(1)L(1) ≈\approx 0.366004+0.00117338i0.366004 + 0.00117338i
L(12)L(\frac12) ≈\approx 0.366004+0.00117338i0.366004 + 0.00117338i
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)
bad2 1+(0.366+1.36i)T 1 + (0.366 + 1.36i)T
3 1 1
13 1+(2.59+2.5i)T 1 + (2.59 + 2.5i)T
good5 1+3.73T+5T2 1 + 3.73T + 5T^{2}
7 1+(3+1.73i)T+(3.5+6.06i)T2 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2}
11 1+(1+1.73i)T+(−5.5+9.52i)T2 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2}
17 1+(0.232−0.401i)T+(−8.5−14.7i)T2 1 + (0.232 - 0.401i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.633+1.09i)T+(−9.5−16.4i)T2 1 + (-0.633 + 1.09i)T + (-9.5 - 16.4i)T^{2}
23 1+(−4.09−7.09i)T+(−11.5+19.9i)T2 1 + (-4.09 - 7.09i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.59−1.5i)T+(14.5−25.1i)T2 1 + (2.59 - 1.5i)T + (14.5 - 25.1i)T^{2}
31 1−4.73iT−31T2 1 - 4.73iT - 31T^{2}
37 1+(−2.13−3.69i)T+(−18.5+32.0i)T2 1 + (-2.13 - 3.69i)T + (-18.5 + 32.0i)T^{2}
41 1+(−7.96+4.59i)T+(20.5−35.5i)T2 1 + (-7.96 + 4.59i)T + (20.5 - 35.5i)T^{2}
43 1+(−2.19−1.26i)T+(21.5+37.2i)T2 1 + (-2.19 - 1.26i)T + (21.5 + 37.2i)T^{2}
47 1+6.73iT−47T2 1 + 6.73iT - 47T^{2}
53 1−3.92iT−53T2 1 - 3.92iT - 53T^{2}
59 1+(0.267−0.464i)T+(−29.5−51.0i)T2 1 + (0.267 - 0.464i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.866−0.5i)T+(30.5+52.8i)T2 1 + (-0.866 - 0.5i)T + (30.5 + 52.8i)T^{2}
67 1+(−3.63−6.29i)T+(−33.5+58.0i)T2 1 + (-3.63 - 6.29i)T + (-33.5 + 58.0i)T^{2}
71 1+(8.02+4.63i)T+(35.5+61.4i)T2 1 + (8.02 + 4.63i)T + (35.5 + 61.4i)T^{2}
73 1+1.73iT−73T2 1 + 1.73iT - 73T^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1+1.46T+83T2 1 + 1.46T + 83T^{2}
89 1+(6.46−3.73i)T+(44.5−77.0i)T2 1 + (6.46 - 3.73i)T + (44.5 - 77.0i)T^{2}
97 1+(5.19+3i)T+(48.5+84.0i)T2 1 + (5.19 + 3i)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

−10.20989664286557230903905026898, −9.330917448693990929191878802923, −8.478940547346432152694693246186, −7.53280313641319794177888645664, −7.14932147757115709517479286351, −5.45471719228046036740070047748, −4.35373340253496416742980542653, −3.45167755170296340760670696413, −2.98344056565803552958265904600, −0.78726659694127066295922914397, 0.29352901180768094508545508622, 2.75267072976103626288913748477, 4.06390299523671024597746831004, 4.68556433394280365489396628863, 5.93555087171879829135817763144, 6.86694200402559778225905857166, 7.46062638125285613871596881402, 8.236460205087720361808172440822, 9.148060471885340380039605584588, 9.704611788325397638888756522272

Graph of the ZZ-function along the critical line