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Properties

Label 2-1323-1.1-c3-0-13
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  − 3.36·2-s + 3.31·4-s − 2.29·5-s + 15.7·8-s + 7.73·10-s − 44.4·11-s − 20.7·13-s − 79.5·16-s − 13.2·17-s − 62.0·19-s − 7.61·20-s + 149.·22-s + 90.1·23-s − 119.·25-s + 69.8·26-s − 100.·29-s − 94.5·31-s + 141.·32-s + 44.6·34-s + 196.·37-s + 208.·38-s − 36.2·40-s + 253.·41-s − 111.·43-s − 147.·44-s − 303.·46-s + 67.7·47-s + ⋯
L(s)  = 1  − 1.18·2-s + 0.414·4-s − 0.205·5-s + 0.696·8-s + 0.244·10-s − 1.21·11-s − 0.442·13-s − 1.24·16-s − 0.189·17-s − 0.749·19-s − 0.0851·20-s + 1.44·22-s + 0.817·23-s − 0.957·25-s + 0.526·26-s − 0.645·29-s − 0.547·31-s + 0.781·32-s + 0.225·34-s + 0.873·37-s + 0.890·38-s − 0.143·40-s + 0.964·41-s − 0.393·43-s − 0.504·44-s − 0.972·46-s + 0.210·47-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 0.40091231390.4009123139
L(12)L(\frac12) ≈\approx 0.40091231390.4009123139
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+3.36T+8T2 1 + 3.36T + 8T^{2}
5 1+2.29T+125T2 1 + 2.29T + 125T^{2}
11 1+44.4T+1.33e3T2 1 + 44.4T + 1.33e3T^{2}
13 1+20.7T+2.19e3T2 1 + 20.7T + 2.19e3T^{2}
17 1+13.2T+4.91e3T2 1 + 13.2T + 4.91e3T^{2}
19 1+62.0T+6.85e3T2 1 + 62.0T + 6.85e3T^{2}
23 1−90.1T+1.21e4T2 1 - 90.1T + 1.21e4T^{2}
29 1+100.T+2.43e4T2 1 + 100.T + 2.43e4T^{2}
31 1+94.5T+2.97e4T2 1 + 94.5T + 2.97e4T^{2}
37 1−196.T+5.06e4T2 1 - 196.T + 5.06e4T^{2}
41 1−253.T+6.89e4T2 1 - 253.T + 6.89e4T^{2}
43 1+111.T+7.95e4T2 1 + 111.T + 7.95e4T^{2}
47 1−67.7T+1.03e5T2 1 - 67.7T + 1.03e5T^{2}
53 1+358.T+1.48e5T2 1 + 358.T + 1.48e5T^{2}
59 1−92.0T+2.05e5T2 1 - 92.0T + 2.05e5T^{2}
61 1+432.T+2.26e5T2 1 + 432.T + 2.26e5T^{2}
67 1+695.T+3.00e5T2 1 + 695.T + 3.00e5T^{2}
71 1−20.1T+3.57e5T2 1 - 20.1T + 3.57e5T^{2}
73 1+1.09e3T+3.89e5T2 1 + 1.09e3T + 3.89e5T^{2}
79 1−871.T+4.93e5T2 1 - 871.T + 4.93e5T^{2}
83 1−233.T+5.71e5T2 1 - 233.T + 5.71e5T^{2}
89 1−168.T+7.04e5T2 1 - 168.T + 7.04e5T^{2}
97 1+1.14e3T+9.12e5T2 1 + 1.14e3T + 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

−9.279872115648345644102172585744, −8.475649437521100963303264185676, −7.70979845335107404213045358016, −7.28354953906816882142880311915, −6.06939929573574391029796091567, −5.01672629678486810696361036025, −4.17471287823295620746720732657, −2.77160027493984575583104685336, −1.76112432882888312823824299991, −0.37375584359369891671941513857, 0.37375584359369891671941513857, 1.76112432882888312823824299991, 2.77160027493984575583104685336, 4.17471287823295620746720732657, 5.01672629678486810696361036025, 6.06939929573574391029796091567, 7.28354953906816882142880311915, 7.70979845335107404213045358016, 8.475649437521100963303264185676, 9.279872115648345644102172585744

Graph of the ZZ-function along the critical line