Properties

Label 2-538-1.1-c7-0-97
Degree 22
Conductor 538538
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 64.1·3-s + 64·4-s − 288.·5-s + 513.·6-s + 1.13e3·7-s + 512·8-s + 1.92e3·9-s − 2.30e3·10-s + 6.91e3·11-s + 4.10e3·12-s + 8.35e3·13-s + 9.09e3·14-s − 1.84e4·15-s + 4.09e3·16-s − 1.60e4·17-s + 1.54e4·18-s + 1.85e4·19-s − 1.84e4·20-s + 7.29e4·21-s + 5.53e4·22-s + 3.41e4·23-s + 3.28e4·24-s + 4.98e3·25-s + 6.68e4·26-s − 1.67e4·27-s + 7.27e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.37·3-s + 0.5·4-s − 1.03·5-s + 0.969·6-s + 1.25·7-s + 0.353·8-s + 0.880·9-s − 0.729·10-s + 1.56·11-s + 0.685·12-s + 1.05·13-s + 0.885·14-s − 1.41·15-s + 0.250·16-s − 0.791·17-s + 0.622·18-s + 0.619·19-s − 0.515·20-s + 1.71·21-s + 1.10·22-s + 0.585·23-s + 0.484·24-s + 0.0637·25-s + 0.745·26-s − 0.163·27-s + 0.626·28-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: 11
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: 00
Selberg data: (2, 538, ( :7/2), 1)(2,\ 538,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) \approx 7.4586328027.458632802
L(12)L(\frac12) \approx 7.4586328027.458632802
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 18T 1 - 8T
269 11.94e7T 1 - 1.94e7T
good3 164.1T+2.18e3T2 1 - 64.1T + 2.18e3T^{2}
5 1+288.T+7.81e4T2 1 + 288.T + 7.81e4T^{2}
7 11.13e3T+8.23e5T2 1 - 1.13e3T + 8.23e5T^{2}
11 16.91e3T+1.94e7T2 1 - 6.91e3T + 1.94e7T^{2}
13 18.35e3T+6.27e7T2 1 - 8.35e3T + 6.27e7T^{2}
17 1+1.60e4T+4.10e8T2 1 + 1.60e4T + 4.10e8T^{2}
19 11.85e4T+8.93e8T2 1 - 1.85e4T + 8.93e8T^{2}
23 13.41e4T+3.40e9T2 1 - 3.41e4T + 3.40e9T^{2}
29 1+3.80e4T+1.72e10T2 1 + 3.80e4T + 1.72e10T^{2}
31 1+9.23e4T+2.75e10T2 1 + 9.23e4T + 2.75e10T^{2}
37 14.00e5T+9.49e10T2 1 - 4.00e5T + 9.49e10T^{2}
41 1+1.63e5T+1.94e11T2 1 + 1.63e5T + 1.94e11T^{2}
43 16.23e5T+2.71e11T2 1 - 6.23e5T + 2.71e11T^{2}
47 1+1.64e5T+5.06e11T2 1 + 1.64e5T + 5.06e11T^{2}
53 18.51e5T+1.17e12T2 1 - 8.51e5T + 1.17e12T^{2}
59 1+5.90e5T+2.48e12T2 1 + 5.90e5T + 2.48e12T^{2}
61 1+2.22e6T+3.14e12T2 1 + 2.22e6T + 3.14e12T^{2}
67 1+3.77e6T+6.06e12T2 1 + 3.77e6T + 6.06e12T^{2}
71 1+6.37e5T+9.09e12T2 1 + 6.37e5T + 9.09e12T^{2}
73 12.41e6T+1.10e13T2 1 - 2.41e6T + 1.10e13T^{2}
79 1+2.94e6T+1.92e13T2 1 + 2.94e6T + 1.92e13T^{2}
83 19.83e6T+2.71e13T2 1 - 9.83e6T + 2.71e13T^{2}
89 14.92e6T+4.42e13T2 1 - 4.92e6T + 4.42e13T^{2}
97 1+1.81e6T+8.07e13T2 1 + 1.81e6T + 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

−9.290604336243382862236249389259, −8.720134752356523361398541740424, −7.88577939933376008150199466666, −7.25485397904000869844688870902, −6.04652151754761743201002704101, −4.56869412541408041506587890174, −3.96009925249341220674984366973, −3.25016370829111510950547266462, −1.96933230367495089676940005686, −1.09254001688510598134945952326, 1.09254001688510598134945952326, 1.96933230367495089676940005686, 3.25016370829111510950547266462, 3.96009925249341220674984366973, 4.56869412541408041506587890174, 6.04652151754761743201002704101, 7.25485397904000869844688870902, 7.88577939933376008150199466666, 8.720134752356523361398541740424, 9.290604336243382862236249389259

Graph of the ZZ-function along the critical line