Properties

Label 2-538-1.1-c7-0-39
Degree 22
Conductor 538538
Sign 1-1
Analytic cond. 168.063168.063
Root an. cond. 12.963912.9639
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  − 8·2-s − 19.6·3-s + 64·4-s − 242.·5-s + 157.·6-s − 1.44e3·7-s − 512·8-s − 1.80e3·9-s + 1.93e3·10-s − 2.21e3·11-s − 1.25e3·12-s − 1.40e4·13-s + 1.15e4·14-s + 4.76e3·15-s + 4.09e3·16-s − 1.99e4·17-s + 1.44e4·18-s + 3.07e4·19-s − 1.55e4·20-s + 2.83e4·21-s + 1.77e4·22-s + 6.63e4·23-s + 1.00e4·24-s − 1.94e4·25-s + 1.12e5·26-s + 7.84e4·27-s − 9.22e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.420·3-s + 0.5·4-s − 0.866·5-s + 0.297·6-s − 1.58·7-s − 0.353·8-s − 0.823·9-s + 0.612·10-s − 0.502·11-s − 0.210·12-s − 1.77·13-s + 1.12·14-s + 0.364·15-s + 0.250·16-s − 0.986·17-s + 0.582·18-s + 1.02·19-s − 0.433·20-s + 0.667·21-s + 0.355·22-s + 1.13·23-s + 0.148·24-s − 0.248·25-s + 1.25·26-s + 0.766·27-s − 0.794·28-s + ⋯

Functional equation

Λ(s)=(538s/2ΓC(s)L(s)=(Λ(8s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(538s/2ΓC(s+7/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    22692 \cdot 269
Sign: 1-1
Analytic conductor: 168.063168.063
Root analytic conductor: 12.963912.9639
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 538, ( :7/2), 1)(2,\ 538,\ (\ :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(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+8T 1 + 8T
269 11.94e7T 1 - 1.94e7T
good3 1+19.6T+2.18e3T2 1 + 19.6T + 2.18e3T^{2}
5 1+242.T+7.81e4T2 1 + 242.T + 7.81e4T^{2}
7 1+1.44e3T+8.23e5T2 1 + 1.44e3T + 8.23e5T^{2}
11 1+2.21e3T+1.94e7T2 1 + 2.21e3T + 1.94e7T^{2}
13 1+1.40e4T+6.27e7T2 1 + 1.40e4T + 6.27e7T^{2}
17 1+1.99e4T+4.10e8T2 1 + 1.99e4T + 4.10e8T^{2}
19 13.07e4T+8.93e8T2 1 - 3.07e4T + 8.93e8T^{2}
23 16.63e4T+3.40e9T2 1 - 6.63e4T + 3.40e9T^{2}
29 12.43e5T+1.72e10T2 1 - 2.43e5T + 1.72e10T^{2}
31 15.66e4T+2.75e10T2 1 - 5.66e4T + 2.75e10T^{2}
37 1+5.31e5T+9.49e10T2 1 + 5.31e5T + 9.49e10T^{2}
41 13.59e5T+1.94e11T2 1 - 3.59e5T + 1.94e11T^{2}
43 15.52e5T+2.71e11T2 1 - 5.52e5T + 2.71e11T^{2}
47 1+2.99e5T+5.06e11T2 1 + 2.99e5T + 5.06e11T^{2}
53 12.10e5T+1.17e12T2 1 - 2.10e5T + 1.17e12T^{2}
59 12.94e6T+2.48e12T2 1 - 2.94e6T + 2.48e12T^{2}
61 1+2.89e6T+3.14e12T2 1 + 2.89e6T + 3.14e12T^{2}
67 11.08e6T+6.06e12T2 1 - 1.08e6T + 6.06e12T^{2}
71 1+4.83e6T+9.09e12T2 1 + 4.83e6T + 9.09e12T^{2}
73 1+4.65e6T+1.10e13T2 1 + 4.65e6T + 1.10e13T^{2}
79 1+6.00e5T+1.92e13T2 1 + 6.00e5T + 1.92e13T^{2}
83 14.21e6T+2.71e13T2 1 - 4.21e6T + 2.71e13T^{2}
89 1+6.58e6T+4.42e13T2 1 + 6.58e6T + 4.42e13T^{2}
97 1+1.36e7T+8.07e13T2 1 + 1.36e7T + 8.07e13T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.297188512502135514457579846562, −8.442917379104110740816191548732, −7.30096373562027807376639611486, −6.82760036111758859431930650546, −5.69562166299809962121910497104, −4.62797919893298041782563168210, −3.11195367033617062222828259717, −2.62889722120193749705310329577, −0.61389306659533056323248079613, 0, 0.61389306659533056323248079613, 2.62889722120193749705310329577, 3.11195367033617062222828259717, 4.62797919893298041782563168210, 5.69562166299809962121910497104, 6.82760036111758859431930650546, 7.30096373562027807376639611486, 8.442917379104110740816191548732, 9.297188512502135514457579846562

Graph of the ZZ-function along the critical line