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Properties

Label 2-1323-1.1-c3-0-75
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  + 5.15·2-s + 18.5·4-s − 17.0·5-s + 54.6·8-s − 87.7·10-s − 30.3·11-s + 89.5·13-s + 132.·16-s − 13.4·17-s + 5.37·19-s − 316.·20-s − 156.·22-s + 167.·23-s + 164.·25-s + 461.·26-s − 135.·29-s + 18.9·31-s + 248.·32-s − 69.1·34-s + 402.·37-s + 27.7·38-s − 929.·40-s + 434.·41-s + 53.1·43-s − 564.·44-s + 863.·46-s + 155.·47-s + ⋯
L(s)  = 1  + 1.82·2-s + 2.32·4-s − 1.52·5-s + 2.41·8-s − 2.77·10-s − 0.831·11-s + 1.91·13-s + 2.07·16-s − 0.191·17-s + 0.0648·19-s − 3.53·20-s − 1.51·22-s + 1.51·23-s + 1.31·25-s + 3.48·26-s − 0.864·29-s + 0.109·31-s + 1.37·32-s − 0.348·34-s + 1.78·37-s + 0.118·38-s − 3.67·40-s + 1.65·41-s + 0.188·43-s − 1.93·44-s + 2.76·46-s + 0.481·47-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 5.7693537225.769353722
L(12)L(\frac12) ≈\approx 5.7693537225.769353722
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 1−5.15T+8T2 1 - 5.15T + 8T^{2}
5 1+17.0T+125T2 1 + 17.0T + 125T^{2}
11 1+30.3T+1.33e3T2 1 + 30.3T + 1.33e3T^{2}
13 1−89.5T+2.19e3T2 1 - 89.5T + 2.19e3T^{2}
17 1+13.4T+4.91e3T2 1 + 13.4T + 4.91e3T^{2}
19 1−5.37T+6.85e3T2 1 - 5.37T + 6.85e3T^{2}
23 1−167.T+1.21e4T2 1 - 167.T + 1.21e4T^{2}
29 1+135.T+2.43e4T2 1 + 135.T + 2.43e4T^{2}
31 1−18.9T+2.97e4T2 1 - 18.9T + 2.97e4T^{2}
37 1−402.T+5.06e4T2 1 - 402.T + 5.06e4T^{2}
41 1−434.T+6.89e4T2 1 - 434.T + 6.89e4T^{2}
43 1−53.1T+7.95e4T2 1 - 53.1T + 7.95e4T^{2}
47 1−155.T+1.03e5T2 1 - 155.T + 1.03e5T^{2}
53 1+301.T+1.48e5T2 1 + 301.T + 1.48e5T^{2}
59 1+412.T+2.05e5T2 1 + 412.T + 2.05e5T^{2}
61 1−571.T+2.26e5T2 1 - 571.T + 2.26e5T^{2}
67 1−820.T+3.00e5T2 1 - 820.T + 3.00e5T^{2}
71 1+8.95T+3.57e5T2 1 + 8.95T + 3.57e5T^{2}
73 1+21.9T+3.89e5T2 1 + 21.9T + 3.89e5T^{2}
79 1−619.T+4.93e5T2 1 - 619.T + 4.93e5T^{2}
83 1−259.T+5.71e5T2 1 - 259.T + 5.71e5T^{2}
89 1−484.T+7.04e5T2 1 - 484.T + 7.04e5T^{2}
97 1−252.T+9.12e5T2 1 - 252.T + 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

−9.113912978028741830044018854135, −8.047430028595814222394434173126, −7.50174896149147874827368716344, −6.56218254597625792967669205816, −5.77152848922439391695419823672, −4.83907158347323345339405052832, −4.03628000803736694765882855051, −3.47786240323766936653072639440, −2.58520971911799224825745820198, −0.941349331093177547140292894585, 0.941349331093177547140292894585, 2.58520971911799224825745820198, 3.47786240323766936653072639440, 4.03628000803736694765882855051, 4.83907158347323345339405052832, 5.77152848922439391695419823672, 6.56218254597625792967669205816, 7.50174896149147874827368716344, 8.047430028595814222394434173126, 9.113912978028741830044018854135

Graph of the ZZ-function along the critical line