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Properties

Label 2-21e2-1.1-c5-0-61
Degree 22
Conductor 441441
Sign 11
Analytic cond. 70.729270.7292
Root an. cond. 8.410068.41006
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  + 8.71·2-s + 44.0·4-s + 99.6·5-s + 104.·8-s + 868.·10-s + 374.·11-s + 868.·13-s − 495.·16-s − 1.09e3·17-s − 868.·19-s + 4.38e3·20-s + 3.26e3·22-s + 2.97e3·23-s + 6.81e3·25-s + 7.57e3·26-s + 4.51e3·29-s − 9.55e3·31-s − 7.67e3·32-s − 9.55e3·34-s − 5.46e3·37-s − 7.57e3·38-s + 1.04e4·40-s + 9.86e3·41-s + 1.25e4·43-s + 1.64e4·44-s + 2.59e4·46-s − 9.96e3·47-s + ⋯
L(s)  = 1  + 1.54·2-s + 1.37·4-s + 1.78·5-s + 0.577·8-s + 2.74·10-s + 0.934·11-s + 1.42·13-s − 0.484·16-s − 0.920·17-s − 0.552·19-s + 2.45·20-s + 1.43·22-s + 1.17·23-s + 2.17·25-s + 2.19·26-s + 0.997·29-s − 1.78·31-s − 1.32·32-s − 1.41·34-s − 0.656·37-s − 0.851·38-s + 1.03·40-s + 0.916·41-s + 1.03·43-s + 1.28·44-s + 1.80·46-s − 0.658·47-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 70.729270.7292
Root analytic conductor: 8.410068.41006
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 441, ( :5/2), 1)(2,\ 441,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 8.0325123318.032512331
L(12)L(\frac12) ≈\approx 8.0325123318.032512331
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
7 1 1
good2 1−8.71T+32T2 1 - 8.71T + 32T^{2}
5 1−99.6T+3.12e3T2 1 - 99.6T + 3.12e3T^{2}
11 1−374.T+1.61e5T2 1 - 374.T + 1.61e5T^{2}
13 1−868.T+3.71e5T2 1 - 868.T + 3.71e5T^{2}
17 1+1.09e3T+1.41e6T2 1 + 1.09e3T + 1.41e6T^{2}
19 1+868.T+2.47e6T2 1 + 868.T + 2.47e6T^{2}
23 1−2.97e3T+6.43e6T2 1 - 2.97e3T + 6.43e6T^{2}
29 1−4.51e3T+2.05e7T2 1 - 4.51e3T + 2.05e7T^{2}
31 1+9.55e3T+2.86e7T2 1 + 9.55e3T + 2.86e7T^{2}
37 1+5.46e3T+6.93e7T2 1 + 5.46e3T + 6.93e7T^{2}
41 1−9.86e3T+1.15e8T2 1 - 9.86e3T + 1.15e8T^{2}
43 1−1.25e4T+1.47e8T2 1 - 1.25e4T + 1.47e8T^{2}
47 1+9.96e3T+2.29e8T2 1 + 9.96e3T + 2.29e8T^{2}
53 1+1.51e4T+4.18e8T2 1 + 1.51e4T + 4.18e8T^{2}
59 1−4.26e4T+7.14e8T2 1 - 4.26e4T + 7.14e8T^{2}
61 1+4.77e4T+8.44e8T2 1 + 4.77e4T + 8.44e8T^{2}
67 1+2.99e4T+1.35e9T2 1 + 2.99e4T + 1.35e9T^{2}
71 1−6.14e4T+1.80e9T2 1 - 6.14e4T + 1.80e9T^{2}
73 1+4.86e4T+2.07e9T2 1 + 4.86e4T + 2.07e9T^{2}
79 1−8.01e4T+3.07e9T2 1 - 8.01e4T + 3.07e9T^{2}
83 1−3.07e4T+3.93e9T2 1 - 3.07e4T + 3.93e9T^{2}
89 1+2.08e4T+5.58e9T2 1 + 2.08e4T + 5.58e9T^{2}
97 1+1.33e5T+8.58e9T2 1 + 1.33e5T + 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

−10.70082444135937252094094836590, −9.260190183982918571532015724875, −8.854998182208820853177482868653, −6.82587963752568901565701215656, −6.29512335163101504674777760737, −5.60087651655389706781391121257, −4.60144330627534084647586392951, −3.50545723784076952286868138194, −2.33924826763764349662170238856, −1.35622425856082421627766480521, 1.35622425856082421627766480521, 2.33924826763764349662170238856, 3.50545723784076952286868138194, 4.60144330627534084647586392951, 5.60087651655389706781391121257, 6.29512335163101504674777760737, 6.82587963752568901565701215656, 8.854998182208820853177482868653, 9.260190183982918571532015724875, 10.70082444135937252094094836590

Graph of the ZZ-function along the critical line