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Properties

Label 2-1323-1.1-c3-0-149
Degree 22
Conductor 13231323
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.62·2-s + 5.17·4-s + 4.84·5-s − 10.2·8-s + 17.5·10-s + 20.1·11-s − 72.2·13-s − 78.6·16-s + 132.·17-s − 76.9·19-s + 25.0·20-s + 73.0·22-s + 22.4·23-s − 101.·25-s − 262.·26-s − 193.·29-s − 89.7·31-s − 203.·32-s + 480.·34-s + 47.9·37-s − 279.·38-s − 49.6·40-s + 3.41·41-s − 168.·43-s + 104.·44-s + 81.5·46-s + 163.·47-s + ⋯
L(s)  = 1  + 1.28·2-s + 0.646·4-s + 0.433·5-s − 0.453·8-s + 0.555·10-s + 0.551·11-s − 1.54·13-s − 1.22·16-s + 1.88·17-s − 0.928·19-s + 0.280·20-s + 0.708·22-s + 0.203·23-s − 0.812·25-s − 1.97·26-s − 1.23·29-s − 0.520·31-s − 1.12·32-s + 2.42·34-s + 0.212·37-s − 1.19·38-s − 0.196·40-s + 0.0130·41-s − 0.597·43-s + 0.356·44-s + 0.261·46-s + 0.506·47-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: −1-1
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: 11
Selberg data: (2, 1323, ( :3/2), −1)(2,\ 1323,\ (\ :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)
bad3 1 1
7 1 1
good2 1−3.62T+8T2 1 - 3.62T + 8T^{2}
5 1−4.84T+125T2 1 - 4.84T + 125T^{2}
11 1−20.1T+1.33e3T2 1 - 20.1T + 1.33e3T^{2}
13 1+72.2T+2.19e3T2 1 + 72.2T + 2.19e3T^{2}
17 1−132.T+4.91e3T2 1 - 132.T + 4.91e3T^{2}
19 1+76.9T+6.85e3T2 1 + 76.9T + 6.85e3T^{2}
23 1−22.4T+1.21e4T2 1 - 22.4T + 1.21e4T^{2}
29 1+193.T+2.43e4T2 1 + 193.T + 2.43e4T^{2}
31 1+89.7T+2.97e4T2 1 + 89.7T + 2.97e4T^{2}
37 1−47.9T+5.06e4T2 1 - 47.9T + 5.06e4T^{2}
41 1−3.41T+6.89e4T2 1 - 3.41T + 6.89e4T^{2}
43 1+168.T+7.95e4T2 1 + 168.T + 7.95e4T^{2}
47 1−163.T+1.03e5T2 1 - 163.T + 1.03e5T^{2}
53 1+337.T+1.48e5T2 1 + 337.T + 1.48e5T^{2}
59 1−517.T+2.05e5T2 1 - 517.T + 2.05e5T^{2}
61 1+424.T+2.26e5T2 1 + 424.T + 2.26e5T^{2}
67 1+978.T+3.00e5T2 1 + 978.T + 3.00e5T^{2}
71 1−40.4T+3.57e5T2 1 - 40.4T + 3.57e5T^{2}
73 1+482.T+3.89e5T2 1 + 482.T + 3.89e5T^{2}
79 1+1.07e3T+4.93e5T2 1 + 1.07e3T + 4.93e5T^{2}
83 1+811.T+5.71e5T2 1 + 811.T + 5.71e5T^{2}
89 1−997.T+7.04e5T2 1 - 997.T + 7.04e5T^{2}
97 1−1.45e3T+9.12e5T2 1 - 1.45e3T + 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

−9.042430340079441333087672076475, −7.79854130099281383584608851255, −7.05454601753816632156780003639, −5.98730234035718768595715272733, −5.48678772481665136108743758097, −4.60554978977674733573807022600, −3.73157410514654410789081414592, −2.81489485382786384081831546026, −1.74056630075626407783961112073, 0, 1.74056630075626407783961112073, 2.81489485382786384081831546026, 3.73157410514654410789081414592, 4.60554978977674733573807022600, 5.48678772481665136108743758097, 5.98730234035718768595715272733, 7.05454601753816632156780003639, 7.79854130099281383584608851255, 9.042430340079441333087672076475

Graph of the ZZ-function along the critical line