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Properties

Label 2-33e2-1.1-c5-0-11
Degree 22
Conductor 10891089
Sign 11
Analytic cond. 174.657174.657
Root an. cond. 13.215813.2158
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.51·2-s − 19.6·4-s − 88.3·5-s − 157.·7-s + 181.·8-s + 311.·10-s + 700.·13-s + 553.·14-s − 11.8·16-s − 831.·17-s + 764.·19-s + 1.73e3·20-s − 1.02e3·23-s + 4.68e3·25-s − 2.46e3·26-s + 3.08e3·28-s + 902.·29-s − 7.63e3·31-s − 5.77e3·32-s + 2.92e3·34-s + 1.39e4·35-s − 1.38e4·37-s − 2.69e3·38-s − 1.60e4·40-s + 1.74e4·41-s + 4.26e3·43-s + 3.62e3·46-s + ⋯
L(s)  = 1  − 0.622·2-s − 0.612·4-s − 1.58·5-s − 1.21·7-s + 1.00·8-s + 0.983·10-s + 1.14·13-s + 0.755·14-s − 0.0115·16-s − 0.697·17-s + 0.486·19-s + 0.968·20-s − 0.405·23-s + 1.49·25-s − 0.715·26-s + 0.743·28-s + 0.199·29-s − 1.42·31-s − 0.996·32-s + 0.434·34-s + 1.91·35-s − 1.66·37-s − 0.302·38-s − 1.58·40-s + 1.61·41-s + 0.351·43-s + 0.252·46-s + ⋯

Functional equation

Λ(s)=(1089s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(1089s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10891089    =    32⋅1123^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 174.657174.657
Root analytic conductor: 13.215813.2158
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1089, ( :5/2), 1)(2,\ 1089,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.068379623460.06837962346
L(12)L(\frac12) ≈\approx 0.068379623460.06837962346
L(72)L(\frac{7}{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)
bad3 1 1
11 1 1
good2 1+3.51T+32T2 1 + 3.51T + 32T^{2}
5 1+88.3T+3.12e3T2 1 + 88.3T + 3.12e3T^{2}
7 1+157.T+1.68e4T2 1 + 157.T + 1.68e4T^{2}
13 1−700.T+3.71e5T2 1 - 700.T + 3.71e5T^{2}
17 1+831.T+1.41e6T2 1 + 831.T + 1.41e6T^{2}
19 1−764.T+2.47e6T2 1 - 764.T + 2.47e6T^{2}
23 1+1.02e3T+6.43e6T2 1 + 1.02e3T + 6.43e6T^{2}
29 1−902.T+2.05e7T2 1 - 902.T + 2.05e7T^{2}
31 1+7.63e3T+2.86e7T2 1 + 7.63e3T + 2.86e7T^{2}
37 1+1.38e4T+6.93e7T2 1 + 1.38e4T + 6.93e7T^{2}
41 1−1.74e4T+1.15e8T2 1 - 1.74e4T + 1.15e8T^{2}
43 1−4.26e3T+1.47e8T2 1 - 4.26e3T + 1.47e8T^{2}
47 1+1.95e4T+2.29e8T2 1 + 1.95e4T + 2.29e8T^{2}
53 1+8.65e3T+4.18e8T2 1 + 8.65e3T + 4.18e8T^{2}
59 1+5.13e4T+7.14e8T2 1 + 5.13e4T + 7.14e8T^{2}
61 1+3.19e4T+8.44e8T2 1 + 3.19e4T + 8.44e8T^{2}
67 1−3.01e4T+1.35e9T2 1 - 3.01e4T + 1.35e9T^{2}
71 1−3.31e4T+1.80e9T2 1 - 3.31e4T + 1.80e9T^{2}
73 1+4.47e4T+2.07e9T2 1 + 4.47e4T + 2.07e9T^{2}
79 1+7.02e4T+3.07e9T2 1 + 7.02e4T + 3.07e9T^{2}
83 1+9.09e4T+3.93e9T2 1 + 9.09e4T + 3.93e9T^{2}
89 1+1.36e5T+5.58e9T2 1 + 1.36e5T + 5.58e9T^{2}
97 1−1.79e4T+8.58e9T2 1 - 1.79e4T + 8.58e9T^{2}
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   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

−9.024098507232539117402568695687, −8.434703698355867714311864986631, −7.61578402767065846703468917751, −6.91821523043985095741698932152, −5.84329650519419073234541378112, −4.56214347527659237777687779673, −3.80586115735969575280587412819, −3.19252846939715397135654859911, −1.37395676105794566630609640458, −0.13384366472601920504430869747, 0.13384366472601920504430869747, 1.37395676105794566630609640458, 3.19252846939715397135654859911, 3.80586115735969575280587412819, 4.56214347527659237777687779673, 5.84329650519419073234541378112, 6.91821523043985095741698932152, 7.61578402767065846703468917751, 8.434703698355867714311864986631, 9.024098507232539117402568695687

Graph of the ZZ-function along the critical line