↑

Properties

Label 2-192-3.2-c8-0-35
Degree 22
Conductor 192192
Sign 11
Analytic cond. 78.216678.2166
Root an. cond. 8.844028.84402
Motivic weight 88
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 81·3-s + 4.03e3·7-s + 6.56e3·9-s + 3.58e4·13-s + 2.58e5·19-s − 3.26e5·21-s + 3.90e5·25-s − 5.31e5·27-s − 1.80e6·31-s − 5.03e5·37-s − 2.90e6·39-s − 3.49e6·43-s + 1.05e7·49-s − 2.09e7·57-s + 2.38e7·61-s + 2.64e7·63-s + 5.42e6·67-s + 1.61e7·73-s − 3.16e7·75-s − 1.88e7·79-s + 4.30e7·81-s + 1.44e8·91-s + 1.46e8·93-s + 1.76e8·97-s + 4.44e7·103-s − 2.03e8·109-s + 4.07e7·111-s + ⋯
L(s)  = 1  − 3-s + 1.68·7-s + 9-s + 1.25·13-s + 1.98·19-s − 1.68·21-s + 25-s − 27-s − 1.95·31-s − 0.268·37-s − 1.25·39-s − 1.02·43-s + 1.82·49-s − 1.98·57-s + 1.72·61-s + 1.68·63-s + 0.269·67-s + 0.569·73-s − 75-s − 0.484·79-s + 81-s + 2.10·91-s + 1.95·93-s + 1.99·97-s + 0.394·103-s − 1.43·109-s + 0.268·111-s + ⋯

Functional equation

Λ(s)=(192s/2 ΓC(s) L(s)=( Λ(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
Λ(s)=(192s/2 ΓC(s+4) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 192192    =    26⋅32^{6} \cdot 3
Sign: 11
Analytic conductor: 78.216678.2166
Root analytic conductor: 8.844028.84402
Motivic weight: 88
Rational: yes
Arithmetic: yes
Character: χ192(65,⋅)\chi_{192} (65, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 192, ( :4), 1)(2,\ 192,\ (\ :4),\ 1)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 2.2947954402.294795440
L(12)L(\frac12) ≈\approx 2.2947954402.294795440
L(5)L(5) 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 1
3 1+p4T 1 + p^{4} T
good5 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
7 1−4034T+p8T2 1 - 4034 T + p^{8} T^{2}
11 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
13 1−35806T+p8T2 1 - 35806 T + p^{8} T^{2}
17 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
19 1−258526T+p8T2 1 - 258526 T + p^{8} T^{2}
23 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
29 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
31 1+1809406T+p8T2 1 + 1809406 T + p^{8} T^{2}
37 1+503522T+p8T2 1 + 503522 T + p^{8} T^{2}
41 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
43 1+3492194T+p8T2 1 + 3492194 T + p^{8} T^{2}
47 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
53 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
59 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
61 1−23826526T+p8T2 1 - 23826526 T + p^{8} T^{2}
67 1−5421406T+p8T2 1 - 5421406 T + p^{8} T^{2}
71 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
73 1−16169282T+p8T2 1 - 16169282 T + p^{8} T^{2}
79 1+18887038T+p8T2 1 + 18887038 T + p^{8} T^{2}
83 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
89 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
97 1−176908034T+p8T2 1 - 176908034 T + p^{8} 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

−11.28482388997223802256791078736, −10.39223498721639189144440755260, −9.043911697744886252759333097469, −7.912123281311546918097956985348, −6.97214534267064460346636515823, −5.56541434739854537542664646704, −4.98680530962777232607116562015, −3.67859969849509510900533029740, −1.67013562884374873004584515589, −0.906491300799226674814448844546, 0.906491300799226674814448844546, 1.67013562884374873004584515589, 3.67859969849509510900533029740, 4.98680530962777232607116562015, 5.56541434739854537542664646704, 6.97214534267064460346636515823, 7.912123281311546918097956985348, 9.043911697744886252759333097469, 10.39223498721639189144440755260, 11.28482388997223802256791078736

Graph of the ZZ-function along the critical line