↑

Properties

Label 2-4304-1.1-c1-0-12
Degree 22
Conductor 43044304
Sign 11
Analytic cond. 34.367634.3676
Root an. cond. 5.862385.86238
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.801·3-s − 3.55·5-s + 0.142·7-s − 2.35·9-s − 5.09·11-s − 3.44·13-s − 2.84·15-s + 5.28·17-s − 4.97·19-s + 0.113·21-s − 1.58·23-s + 7.62·25-s − 4.29·27-s + 5.53·29-s + 0.382·31-s − 4.08·33-s − 0.504·35-s − 6.21·37-s − 2.76·39-s − 11.4·41-s + 9.71·43-s + 8.37·45-s + 9.64·47-s − 6.97·49-s + 4.23·51-s − 10.7·53-s + 18.1·55-s + ⋯
L(s)  = 1  + 0.462·3-s − 1.58·5-s + 0.0536·7-s − 0.786·9-s − 1.53·11-s − 0.955·13-s − 0.734·15-s + 1.28·17-s − 1.14·19-s + 0.0248·21-s − 0.329·23-s + 1.52·25-s − 0.826·27-s + 1.02·29-s + 0.0686·31-s − 0.710·33-s − 0.0852·35-s − 1.02·37-s − 0.442·39-s − 1.78·41-s + 1.48·43-s + 1.24·45-s + 1.40·47-s − 0.997·49-s + 0.593·51-s − 1.47·53-s + 2.44·55-s + ⋯

Functional equation

Λ(s)=(4304s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4304s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43044304    =    24⋅2692^{4} \cdot 269
Sign: 11
Analytic conductor: 34.367634.3676
Root analytic conductor: 5.862385.86238
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4304, ( :1/2), 1)(2,\ 4304,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.63300733500.6330073350
L(12)L(\frac12) ≈\approx 0.63300733500.6330073350
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 1
269 1−T 1 - T
good3 1−0.801T+3T2 1 - 0.801T + 3T^{2}
5 1+3.55T+5T2 1 + 3.55T + 5T^{2}
7 1−0.142T+7T2 1 - 0.142T + 7T^{2}
11 1+5.09T+11T2 1 + 5.09T + 11T^{2}
13 1+3.44T+13T2 1 + 3.44T + 13T^{2}
17 1−5.28T+17T2 1 - 5.28T + 17T^{2}
19 1+4.97T+19T2 1 + 4.97T + 19T^{2}
23 1+1.58T+23T2 1 + 1.58T + 23T^{2}
29 1−5.53T+29T2 1 - 5.53T + 29T^{2}
31 1−0.382T+31T2 1 - 0.382T + 31T^{2}
37 1+6.21T+37T2 1 + 6.21T + 37T^{2}
41 1+11.4T+41T2 1 + 11.4T + 41T^{2}
43 1−9.71T+43T2 1 - 9.71T + 43T^{2}
47 1−9.64T+47T2 1 - 9.64T + 47T^{2}
53 1+10.7T+53T2 1 + 10.7T + 53T^{2}
59 1+2.10T+59T2 1 + 2.10T + 59T^{2}
61 1−8.92T+61T2 1 - 8.92T + 61T^{2}
67 1+12.4T+67T2 1 + 12.4T + 67T^{2}
71 1−7.78T+71T2 1 - 7.78T + 71T^{2}
73 1−8.61T+73T2 1 - 8.61T + 73T^{2}
79 1−11.8T+79T2 1 - 11.8T + 79T^{2}
83 1+5.61T+83T2 1 + 5.61T + 83T^{2}
89 1+12.7T+89T2 1 + 12.7T + 89T^{2}
97 1−8.41T+97T2 1 - 8.41T + 97T^{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

−8.296129842860430483279152844179, −7.72294591809693113764955626836, −7.32367313503145051316664458925, −6.20671182338768656527290791269, −5.20252625967378560915463979993, −4.67723594421279885402725194247, −3.62867725077883429654692709453, −3.03333700248008130018470830300, −2.22762064301727296837879733045, −0.40821075168406269014565020978, 0.40821075168406269014565020978, 2.22762064301727296837879733045, 3.03333700248008130018470830300, 3.62867725077883429654692709453, 4.67723594421279885402725194247, 5.20252625967378560915463979993, 6.20671182338768656527290791269, 7.32367313503145051316664458925, 7.72294591809693113764955626836, 8.296129842860430483279152844179

Graph of the ZZ-function along the critical line