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Properties

Label 2-1323-1.1-c3-0-119
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  − 0.560·2-s − 7.68·4-s + 12.9·5-s + 8.79·8-s − 7.26·10-s − 48.2·11-s − 36.3·13-s + 56.5·16-s + 83.2·17-s + 67.4·19-s − 99.5·20-s + 27.0·22-s + 30.6·23-s + 42.8·25-s + 20.3·26-s − 294.·29-s + 270.·31-s − 102.·32-s − 46.7·34-s − 204.·37-s − 37.8·38-s + 114.·40-s + 287.·41-s − 55.4·43-s + 370.·44-s − 17.1·46-s − 191.·47-s + ⋯
L(s)  = 1  − 0.198·2-s − 0.960·4-s + 1.15·5-s + 0.388·8-s − 0.229·10-s − 1.32·11-s − 0.774·13-s + 0.883·16-s + 1.18·17-s + 0.814·19-s − 1.11·20-s + 0.262·22-s + 0.277·23-s + 0.342·25-s + 0.153·26-s − 1.88·29-s + 1.56·31-s − 0.564·32-s − 0.235·34-s − 0.907·37-s − 0.161·38-s + 0.450·40-s + 1.09·41-s − 0.196·43-s + 1.26·44-s − 0.0550·46-s − 0.593·47-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+0.560T+8T2 1 + 0.560T + 8T^{2}
5 1−12.9T+125T2 1 - 12.9T + 125T^{2}
11 1+48.2T+1.33e3T2 1 + 48.2T + 1.33e3T^{2}
13 1+36.3T+2.19e3T2 1 + 36.3T + 2.19e3T^{2}
17 1−83.2T+4.91e3T2 1 - 83.2T + 4.91e3T^{2}
19 1−67.4T+6.85e3T2 1 - 67.4T + 6.85e3T^{2}
23 1−30.6T+1.21e4T2 1 - 30.6T + 1.21e4T^{2}
29 1+294.T+2.43e4T2 1 + 294.T + 2.43e4T^{2}
31 1−270.T+2.97e4T2 1 - 270.T + 2.97e4T^{2}
37 1+204.T+5.06e4T2 1 + 204.T + 5.06e4T^{2}
41 1−287.T+6.89e4T2 1 - 287.T + 6.89e4T^{2}
43 1+55.4T+7.95e4T2 1 + 55.4T + 7.95e4T^{2}
47 1+191.T+1.03e5T2 1 + 191.T + 1.03e5T^{2}
53 1+521.T+1.48e5T2 1 + 521.T + 1.48e5T^{2}
59 1+381.T+2.05e5T2 1 + 381.T + 2.05e5T^{2}
61 1−155.T+2.26e5T2 1 - 155.T + 2.26e5T^{2}
67 1+65.1T+3.00e5T2 1 + 65.1T + 3.00e5T^{2}
71 1−256.T+3.57e5T2 1 - 256.T + 3.57e5T^{2}
73 1−318.T+3.89e5T2 1 - 318.T + 3.89e5T^{2}
79 1−77.7T+4.93e5T2 1 - 77.7T + 4.93e5T^{2}
83 1−836.T+5.71e5T2 1 - 836.T + 5.71e5T^{2}
89 1−1.59e3T+7.04e5T2 1 - 1.59e3T + 7.04e5T^{2}
97 1+1.18e3T+9.12e5T2 1 + 1.18e3T + 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

−9.099134005325966548476607023407, −7.921615507474618230481853505224, −7.56631522414069298780923564763, −6.15713909348922837863888638722, −5.27189916909220284715486125331, −4.98638985031370177412622639434, −3.52630624393083522607441266943, −2.49788349514918925902341456587, −1.28865727498584934176123859257, 0, 1.28865727498584934176123859257, 2.49788349514918925902341456587, 3.52630624393083522607441266943, 4.98638985031370177412622639434, 5.27189916909220284715486125331, 6.15713909348922837863888638722, 7.56631522414069298780923564763, 7.921615507474618230481853505224, 9.099134005325966548476607023407

Graph of the ZZ-function along the critical line