↑

Properties

Label 2-2496-1.1-c3-0-104
Degree 22
Conductor 24962496
Sign −1-1
Analytic cond. 147.268147.268
Root an. cond. 12.135412.1354
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 7.11·5-s + 8.88·7-s + 9·9-s − 26·11-s + 13·13-s − 21.3·15-s + 16.2·17-s − 35.5·19-s − 26.6·21-s + 153.·23-s − 74.3·25-s − 27·27-s − 223.·29-s + 126.·31-s + 78·33-s + 63.2·35-s − 217.·37-s − 39·39-s + 105.·41-s + 183.·43-s + 64.0·45-s + 96.6·47-s − 264.·49-s − 48.6·51-s − 386.·53-s − 184.·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.636·5-s + 0.479·7-s + 0.333·9-s − 0.712·11-s + 0.277·13-s − 0.367·15-s + 0.231·17-s − 0.429·19-s − 0.276·21-s + 1.39·23-s − 0.595·25-s − 0.192·27-s − 1.43·29-s + 0.734·31-s + 0.411·33-s + 0.305·35-s − 0.966·37-s − 0.160·39-s + 0.403·41-s + 0.651·43-s + 0.212·45-s + 0.300·47-s − 0.769·49-s − 0.133·51-s − 1.00·53-s − 0.453·55-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: −1-1
Analytic conductor: 147.268147.268
Root analytic conductor: 12.135412.1354
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2496, ( :3/2), −1)(2,\ 2496,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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 1
3 1+3T 1 + 3T
13 1−13T 1 - 13T
good5 1−7.11T+125T2 1 - 7.11T + 125T^{2}
7 1−8.88T+343T2 1 - 8.88T + 343T^{2}
11 1+26T+1.33e3T2 1 + 26T + 1.33e3T^{2}
17 1−16.2T+4.91e3T2 1 - 16.2T + 4.91e3T^{2}
19 1+35.5T+6.85e3T2 1 + 35.5T + 6.85e3T^{2}
23 1−153.T+1.21e4T2 1 - 153.T + 1.21e4T^{2}
29 1+223.T+2.43e4T2 1 + 223.T + 2.43e4T^{2}
31 1−126.T+2.97e4T2 1 - 126.T + 2.97e4T^{2}
37 1+217.T+5.06e4T2 1 + 217.T + 5.06e4T^{2}
41 1−105.T+6.89e4T2 1 - 105.T + 6.89e4T^{2}
43 1−183.T+7.95e4T2 1 - 183.T + 7.95e4T^{2}
47 1−96.6T+1.03e5T2 1 - 96.6T + 1.03e5T^{2}
53 1+386.T+1.48e5T2 1 + 386.T + 1.48e5T^{2}
59 1+34.5T+2.05e5T2 1 + 34.5T + 2.05e5T^{2}
61 1+274.T+2.26e5T2 1 + 274.T + 2.26e5T^{2}
67 1+93.9T+3.00e5T2 1 + 93.9T + 3.00e5T^{2}
71 1+741.T+3.57e5T2 1 + 741.T + 3.57e5T^{2}
73 1−640.T+3.89e5T2 1 - 640.T + 3.89e5T^{2}
79 1+182.T+4.93e5T2 1 + 182.T + 4.93e5T^{2}
83 1−288.T+5.71e5T2 1 - 288.T + 5.71e5T^{2}
89 1−963.T+7.04e5T2 1 - 963.T + 7.04e5T^{2}
97 1+481.T+9.12e5T2 1 + 481.T + 9.12e5T^{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

−8.105393538583679537911402683128, −7.42906955836058059761279398717, −6.55110092600893619815788059161, −5.74820733241354185680904445960, −5.18667476648359443372132212885, −4.36969221592711796752941596261, −3.23061536413008528649680397516, −2.13841746579484245052465215851, −1.24811313124139847134225459235, 0, 1.24811313124139847134225459235, 2.13841746579484245052465215851, 3.23061536413008528649680397516, 4.36969221592711796752941596261, 5.18667476648359443372132212885, 5.74820733241354185680904445960, 6.55110092600893619815788059161, 7.42906955836058059761279398717, 8.105393538583679537911402683128

Graph of the ZZ-function along the critical line