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Properties

Label 2-538-1.1-c7-0-148
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 + 70.5·3-s + 64·4-s − 243.·5-s + 564.·6-s + 620.·7-s + 512·8-s + 2.79e3·9-s − 1.95e3·10-s − 5.12e3·11-s + 4.51e3·12-s − 8.49e3·13-s + 4.96e3·14-s − 1.72e4·15-s + 4.09e3·16-s − 3.59e4·17-s + 2.23e4·18-s + 4.69e4·19-s − 1.56e4·20-s + 4.37e4·21-s − 4.10e4·22-s + 5.24e4·23-s + 3.61e4·24-s − 1.86e4·25-s − 6.79e4·26-s + 4.26e4·27-s + 3.97e4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.50·3-s + 0.5·4-s − 0.872·5-s + 1.06·6-s + 0.683·7-s + 0.353·8-s + 1.27·9-s − 0.616·10-s − 1.16·11-s + 0.754·12-s − 1.07·13-s + 0.483·14-s − 1.31·15-s + 0.250·16-s − 1.77·17-s + 0.902·18-s + 1.57·19-s − 0.436·20-s + 1.03·21-s − 0.821·22-s + 0.898·23-s + 0.533·24-s − 0.239·25-s − 0.758·26-s + 0.417·27-s + 0.341·28-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\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    =    2⋅2692 \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(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1−8T 1 - 8T
269 1+1.94e7T 1 + 1.94e7T
good3 1−70.5T+2.18e3T2 1 - 70.5T + 2.18e3T^{2}
5 1+243.T+7.81e4T2 1 + 243.T + 7.81e4T^{2}
7 1−620.T+8.23e5T2 1 - 620.T + 8.23e5T^{2}
11 1+5.12e3T+1.94e7T2 1 + 5.12e3T + 1.94e7T^{2}
13 1+8.49e3T+6.27e7T2 1 + 8.49e3T + 6.27e7T^{2}
17 1+3.59e4T+4.10e8T2 1 + 3.59e4T + 4.10e8T^{2}
19 1−4.69e4T+8.93e8T2 1 - 4.69e4T + 8.93e8T^{2}
23 1−5.24e4T+3.40e9T2 1 - 5.24e4T + 3.40e9T^{2}
29 1−1.60e4T+1.72e10T2 1 - 1.60e4T + 1.72e10T^{2}
31 1−2.29e5T+2.75e10T2 1 - 2.29e5T + 2.75e10T^{2}
37 1+5.17e5T+9.49e10T2 1 + 5.17e5T + 9.49e10T^{2}
41 1−2.76e5T+1.94e11T2 1 - 2.76e5T + 1.94e11T^{2}
43 1+3.47e5T+2.71e11T2 1 + 3.47e5T + 2.71e11T^{2}
47 1+1.35e6T+5.06e11T2 1 + 1.35e6T + 5.06e11T^{2}
53 1+5.36e5T+1.17e12T2 1 + 5.36e5T + 1.17e12T^{2}
59 1+1.09e6T+2.48e12T2 1 + 1.09e6T + 2.48e12T^{2}
61 1+3.00e6T+3.14e12T2 1 + 3.00e6T + 3.14e12T^{2}
67 1+6.42e4T+6.06e12T2 1 + 6.42e4T + 6.06e12T^{2}
71 1−4.09e6T+9.09e12T2 1 - 4.09e6T + 9.09e12T^{2}
73 1+2.03e6T+1.10e13T2 1 + 2.03e6T + 1.10e13T^{2}
79 1+2.18e6T+1.92e13T2 1 + 2.18e6T + 1.92e13T^{2}
83 1−1.77e6T+2.71e13T2 1 - 1.77e6T + 2.71e13T^{2}
89 1+3.68e6T+4.42e13T2 1 + 3.68e6T + 4.42e13T^{2}
97 1−4.07e6T+8.07e13T2 1 - 4.07e6T + 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

−9.073984827298302609742066571022, −8.075910888262557975087845625157, −7.68948848154390743593241102706, −6.82053553856825229717268021827, −5.04042756085922325323650543100, −4.56298799677468563445307215745, −3.30516786757962239324987648422, −2.70891435432081257857649709357, −1.72380756834874292039824640027, 0, 1.72380756834874292039824640027, 2.70891435432081257857649709357, 3.30516786757962239324987648422, 4.56298799677468563445307215745, 5.04042756085922325323650543100, 6.82053553856825229717268021827, 7.68948848154390743593241102706, 8.075910888262557975087845625157, 9.073984827298302609742066571022

Graph of the ZZ-function along the critical line