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Properties

Label 2-522-1.1-c1-0-8
Degree 22
Conductor 522522
Sign −1-1
Analytic cond. 4.168194.16819
Root an. cond. 2.041612.04161
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 3·11-s − 13-s + 2·14-s + 16-s − 8·17-s − 20-s − 3·22-s − 4·23-s − 4·25-s + 26-s − 2·28-s + 29-s − 3·31-s − 32-s + 8·34-s + 2·35-s + 8·37-s + 40-s − 2·41-s − 11·43-s + 3·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.904·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.94·17-s − 0.223·20-s − 0.639·22-s − 0.834·23-s − 4/5·25-s + 0.196·26-s − 0.377·28-s + 0.185·29-s − 0.538·31-s − 0.176·32-s + 1.37·34-s + 0.338·35-s + 1.31·37-s + 0.158·40-s − 0.312·41-s − 1.67·43-s + 0.452·44-s + ⋯

Functional equation

Λ(s)=(522s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(522s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 522522    =    2⋅32⋅292 \cdot 3^{2} \cdot 29
Sign: −1-1
Analytic conductor: 4.168194.16819
Root analytic conductor: 2.041612.04161
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 522, ( :1/2), −1)(2,\ 522,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T 1 + T
3 1 1
29 1−T 1 - T
good5 1+T+pT2 1 + T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+8T+pT2 1 + 8 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+11T+pT2 1 + 11 T + p T^{2}
47 1+13T+pT2 1 + 13 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1+2T+pT2 1 + 2 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−15T+pT2 1 - 15 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+2T+pT2 1 + 2 T + p T^{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

−10.26866147802001152913025958268, −9.482258294877269881590757644436, −8.763894772751692239273117298832, −7.81012507517906630443430989651, −6.74819217569927040729981669784, −6.19818150900186047644659279635, −4.54621693456857955686454044680, −3.46305229707620473692544260201, −2.01064091456439163113006432735, 0, 2.01064091456439163113006432735, 3.46305229707620473692544260201, 4.54621693456857955686454044680, 6.19818150900186047644659279635, 6.74819217569927040729981669784, 7.81012507517906630443430989651, 8.763894772751692239273117298832, 9.482258294877269881590757644436, 10.26866147802001152913025958268

Graph of the ZZ-function along the critical line