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Properties

Label 2-1323-1.1-c3-0-87
Degree 22
Conductor 13231323
Sign 11
Analytic cond. 78.059578.0595
Root an. cond. 8.835138.83513
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.78·2-s − 0.244·4-s + 20.8·5-s − 22.9·8-s + 58.0·10-s + 65.7·11-s − 50.9·13-s − 61.9·16-s − 46.1·17-s + 62.1·19-s − 5.09·20-s + 183.·22-s + 125.·23-s + 308.·25-s − 141.·26-s − 168.·29-s + 187.·31-s + 11.0·32-s − 128.·34-s − 70.0·37-s + 173.·38-s − 478.·40-s + 188.·41-s + 239.·43-s − 16.0·44-s + 349.·46-s + 591.·47-s + ⋯
L(s)  = 1  + 0.984·2-s − 0.0305·4-s + 1.86·5-s − 1.01·8-s + 1.83·10-s + 1.80·11-s − 1.08·13-s − 0.968·16-s − 0.658·17-s + 0.750·19-s − 0.0569·20-s + 1.77·22-s + 1.13·23-s + 2.47·25-s − 1.06·26-s − 1.08·29-s + 1.08·31-s + 0.0611·32-s − 0.647·34-s − 0.311·37-s + 0.738·38-s − 1.89·40-s + 0.717·41-s + 0.849·43-s − 0.0551·44-s + 1.11·46-s + 1.83·47-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 11
Analytic conductor: 78.059578.0595
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1323, ( :3/2), 1)(2,\ 1323,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.0159885015.015988501
L(12)L(\frac12) ≈\approx 5.0159885015.015988501
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)
bad3 1 1
7 1 1
good2 1−2.78T+8T2 1 - 2.78T + 8T^{2}
5 1−20.8T+125T2 1 - 20.8T + 125T^{2}
11 1−65.7T+1.33e3T2 1 - 65.7T + 1.33e3T^{2}
13 1+50.9T+2.19e3T2 1 + 50.9T + 2.19e3T^{2}
17 1+46.1T+4.91e3T2 1 + 46.1T + 4.91e3T^{2}
19 1−62.1T+6.85e3T2 1 - 62.1T + 6.85e3T^{2}
23 1−125.T+1.21e4T2 1 - 125.T + 1.21e4T^{2}
29 1+168.T+2.43e4T2 1 + 168.T + 2.43e4T^{2}
31 1−187.T+2.97e4T2 1 - 187.T + 2.97e4T^{2}
37 1+70.0T+5.06e4T2 1 + 70.0T + 5.06e4T^{2}
41 1−188.T+6.89e4T2 1 - 188.T + 6.89e4T^{2}
43 1−239.T+7.95e4T2 1 - 239.T + 7.95e4T^{2}
47 1−591.T+1.03e5T2 1 - 591.T + 1.03e5T^{2}
53 1+464.T+1.48e5T2 1 + 464.T + 1.48e5T^{2}
59 1+325.T+2.05e5T2 1 + 325.T + 2.05e5T^{2}
61 1+261.T+2.26e5T2 1 + 261.T + 2.26e5T^{2}
67 1−340.T+3.00e5T2 1 - 340.T + 3.00e5T^{2}
71 1+752.T+3.57e5T2 1 + 752.T + 3.57e5T^{2}
73 1−245.T+3.89e5T2 1 - 245.T + 3.89e5T^{2}
79 1−546.T+4.93e5T2 1 - 546.T + 4.93e5T^{2}
83 1+144.T+5.71e5T2 1 + 144.T + 5.71e5T^{2}
89 1+603.T+7.04e5T2 1 + 603.T + 7.04e5T^{2}
97 1−1.35e3T+9.12e5T2 1 - 1.35e3T + 9.12e5T^{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

−9.161832431642491045494564647593, −9.039285798227110830434576582590, −7.25169433135849502073734553840, −6.44337805383435953594737042047, −5.87415210182200023725175113365, −5.04736806075532183373433863722, −4.32014485989271344474754956229, −3.10119958152249523559508703292, −2.19222594412830502245172157774, −1.03140266686704113044661630908, 1.03140266686704113044661630908, 2.19222594412830502245172157774, 3.10119958152249523559508703292, 4.32014485989271344474754956229, 5.04736806075532183373433863722, 5.87415210182200023725175113365, 6.44337805383435953594737042047, 7.25169433135849502073734553840, 9.039285798227110830434576582590, 9.161832431642491045494564647593

Graph of the ZZ-function along the critical line