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Properties

Label 2-384-1.1-c7-0-28
Degree 22
Conductor 384384
Sign 11
Analytic cond. 119.955119.955
Root an. cond. 10.952410.9524
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27·3-s + 514.·5-s + 1.60e3·7-s + 729·9-s − 6.91e3·11-s + 1.23e3·13-s − 1.38e4·15-s + 2.71e4·17-s + 4.41e4·19-s − 4.32e4·21-s − 1.55e4·23-s + 1.86e5·25-s − 1.96e4·27-s − 9.45e4·29-s + 9.79e4·31-s + 1.86e5·33-s + 8.23e5·35-s + 5.56e5·37-s − 3.34e4·39-s − 2.03e5·41-s − 5.95e4·43-s + 3.74e5·45-s − 6.98e5·47-s + 1.73e6·49-s − 7.32e5·51-s − 9.16e4·53-s − 3.55e6·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.84·5-s + 1.76·7-s + 0.333·9-s − 1.56·11-s + 0.156·13-s − 1.06·15-s + 1.33·17-s + 1.47·19-s − 1.01·21-s − 0.265·23-s + 2.38·25-s − 0.192·27-s − 0.720·29-s + 0.590·31-s + 0.904·33-s + 3.24·35-s + 1.80·37-s − 0.0902·39-s − 0.460·41-s − 0.114·43-s + 0.613·45-s − 0.981·47-s + 2.11·49-s − 0.772·51-s − 0.0845·53-s − 2.88·55-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: 11
Analytic conductor: 119.955119.955
Root analytic conductor: 10.952410.9524
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 384, ( :7/2), 1)(2,\ 384,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 3.6878446613.687844661
L(12)L(\frac12) ≈\approx 3.6878446613.687844661
L(92)L(\frac{9}{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)
bad2 1 1
3 1+27T 1 + 27T
good5 1−514.T+7.81e4T2 1 - 514.T + 7.81e4T^{2}
7 1−1.60e3T+8.23e5T2 1 - 1.60e3T + 8.23e5T^{2}
11 1+6.91e3T+1.94e7T2 1 + 6.91e3T + 1.94e7T^{2}
13 1−1.23e3T+6.27e7T2 1 - 1.23e3T + 6.27e7T^{2}
17 1−2.71e4T+4.10e8T2 1 - 2.71e4T + 4.10e8T^{2}
19 1−4.41e4T+8.93e8T2 1 - 4.41e4T + 8.93e8T^{2}
23 1+1.55e4T+3.40e9T2 1 + 1.55e4T + 3.40e9T^{2}
29 1+9.45e4T+1.72e10T2 1 + 9.45e4T + 1.72e10T^{2}
31 1−9.79e4T+2.75e10T2 1 - 9.79e4T + 2.75e10T^{2}
37 1−5.56e5T+9.49e10T2 1 - 5.56e5T + 9.49e10T^{2}
41 1+2.03e5T+1.94e11T2 1 + 2.03e5T + 1.94e11T^{2}
43 1+5.95e4T+2.71e11T2 1 + 5.95e4T + 2.71e11T^{2}
47 1+6.98e5T+5.06e11T2 1 + 6.98e5T + 5.06e11T^{2}
53 1+9.16e4T+1.17e12T2 1 + 9.16e4T + 1.17e12T^{2}
59 1+1.47e6T+2.48e12T2 1 + 1.47e6T + 2.48e12T^{2}
61 1−4.24e4T+3.14e12T2 1 - 4.24e4T + 3.14e12T^{2}
67 1+7.22e5T+6.06e12T2 1 + 7.22e5T + 6.06e12T^{2}
71 1−9.41e5T+9.09e12T2 1 - 9.41e5T + 9.09e12T^{2}
73 1+2.12e6T+1.10e13T2 1 + 2.12e6T + 1.10e13T^{2}
79 1−2.91e6T+1.92e13T2 1 - 2.91e6T + 1.92e13T^{2}
83 1+5.42e6T+2.71e13T2 1 + 5.42e6T + 2.71e13T^{2}
89 1−3.26e6T+4.42e13T2 1 - 3.26e6T + 4.42e13T^{2}
97 1−3.55e6T+8.07e13T2 1 - 3.55e6T + 8.07e13T^{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

−10.17336392269234139810622108345, −9.535452896378048749666869226196, −8.151459508458390675003002948264, −7.48117929704084284921628876282, −5.95369404784908382296830774469, −5.36712953585746690102311733656, −4.84059617212196700263392833672, −2.82648197022283686964580929431, −1.74962375568427336776693155228, −1.01271362503436043365370149704, 1.01271362503436043365370149704, 1.74962375568427336776693155228, 2.82648197022283686964580929431, 4.84059617212196700263392833672, 5.36712953585746690102311733656, 5.95369404784908382296830774469, 7.48117929704084284921628876282, 8.151459508458390675003002948264, 9.535452896378048749666869226196, 10.17336392269234139810622108345

Graph of the ZZ-function along the critical line