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Properties

Label 2-152-1.1-c7-0-26
Degree 22
Conductor 152152
Sign −1-1
Analytic cond. 47.482547.4825
Root an. cond. 6.890756.89075
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 42.3·3-s − 290.·5-s + 951.·7-s − 393.·9-s − 1.38e3·11-s − 1.44e3·13-s − 1.23e4·15-s + 1.74e4·17-s − 6.85e3·19-s + 4.02e4·21-s − 3.83e4·23-s + 6.53e3·25-s − 1.09e5·27-s − 1.82e4·29-s − 1.41e5·31-s − 5.87e4·33-s − 2.76e5·35-s − 7.79e4·37-s − 6.13e4·39-s − 1.81e5·41-s − 3.62e5·43-s + 1.14e5·45-s − 2.80e5·47-s + 8.18e4·49-s + 7.38e5·51-s + 7.00e5·53-s + 4.03e5·55-s + ⋯
L(s)  = 1  + 0.905·3-s − 1.04·5-s + 1.04·7-s − 0.179·9-s − 0.314·11-s − 0.183·13-s − 0.942·15-s + 0.860·17-s − 0.229·19-s + 0.949·21-s − 0.656·23-s + 0.0836·25-s − 1.06·27-s − 0.139·29-s − 0.854·31-s − 0.284·33-s − 1.09·35-s − 0.253·37-s − 0.165·39-s − 0.410·41-s − 0.695·43-s + 0.187·45-s − 0.394·47-s + 0.0993·49-s + 0.779·51-s + 0.646·53-s + 0.327·55-s + ⋯

Functional equation

Λ(s)=(152s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(152s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23⋅192^{3} \cdot 19
Sign: −1-1
Analytic conductor: 47.482547.4825
Root analytic conductor: 6.890756.89075
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 152, ( :7/2), −1)(2,\ 152,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
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
19 1+6.85e3T 1 + 6.85e3T
good3 1−42.3T+2.18e3T2 1 - 42.3T + 2.18e3T^{2}
5 1+290.T+7.81e4T2 1 + 290.T + 7.81e4T^{2}
7 1−951.T+8.23e5T2 1 - 951.T + 8.23e5T^{2}
11 1+1.38e3T+1.94e7T2 1 + 1.38e3T + 1.94e7T^{2}
13 1+1.44e3T+6.27e7T2 1 + 1.44e3T + 6.27e7T^{2}
17 1−1.74e4T+4.10e8T2 1 - 1.74e4T + 4.10e8T^{2}
23 1+3.83e4T+3.40e9T2 1 + 3.83e4T + 3.40e9T^{2}
29 1+1.82e4T+1.72e10T2 1 + 1.82e4T + 1.72e10T^{2}
31 1+1.41e5T+2.75e10T2 1 + 1.41e5T + 2.75e10T^{2}
37 1+7.79e4T+9.49e10T2 1 + 7.79e4T + 9.49e10T^{2}
41 1+1.81e5T+1.94e11T2 1 + 1.81e5T + 1.94e11T^{2}
43 1+3.62e5T+2.71e11T2 1 + 3.62e5T + 2.71e11T^{2}
47 1+2.80e5T+5.06e11T2 1 + 2.80e5T + 5.06e11T^{2}
53 1−7.00e5T+1.17e12T2 1 - 7.00e5T + 1.17e12T^{2}
59 1−7.01e5T+2.48e12T2 1 - 7.01e5T + 2.48e12T^{2}
61 1−1.06e4T+3.14e12T2 1 - 1.06e4T + 3.14e12T^{2}
67 1+6.78e5T+6.06e12T2 1 + 6.78e5T + 6.06e12T^{2}
71 1+3.12e6T+9.09e12T2 1 + 3.12e6T + 9.09e12T^{2}
73 1+6.01e6T+1.10e13T2 1 + 6.01e6T + 1.10e13T^{2}
79 1+7.03e6T+1.92e13T2 1 + 7.03e6T + 1.92e13T^{2}
83 1+2.57e6T+2.71e13T2 1 + 2.57e6T + 2.71e13T^{2}
89 1−2.58e6T+4.42e13T2 1 - 2.58e6T + 4.42e13T^{2}
97 1+6.37e6T+8.07e13T2 1 + 6.37e6T + 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

−11.34796343883995952398940603540, −10.12046283791803100618874282901, −8.748924811800915520017437085823, −8.047044815524387651760357688514, −7.38591532731778299723793199083, −5.53092564911765801729465344731, −4.22578988581419290625039833502, −3.15237535186046106405481313361, −1.76564671840849973152821943236, 0, 1.76564671840849973152821943236, 3.15237535186046106405481313361, 4.22578988581419290625039833502, 5.53092564911765801729465344731, 7.38591532731778299723793199083, 8.047044815524387651760357688514, 8.748924811800915520017437085823, 10.12046283791803100618874282901, 11.34796343883995952398940603540

Graph of the ZZ-function along the critical line