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Properties

Label 2-1638-1.1-c1-0-24
Degree 22
Conductor 16381638
Sign −1-1
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
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 + 7-s − 8-s − 10-s − 5·11-s − 13-s − 14-s + 16-s + 3·17-s − 19-s + 20-s + 5·22-s − 3·23-s − 4·25-s + 26-s + 28-s − 9·29-s + 4·31-s − 32-s − 3·34-s + 35-s − 11·37-s + 38-s − 40-s − 5·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.50·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s + 1.06·22-s − 0.625·23-s − 4/5·25-s + 0.196·26-s + 0.188·28-s − 1.67·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 0.169·35-s − 1.80·37-s + 0.162·38-s − 0.158·40-s − 0.762·43-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: −1-1
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1638, ( :1/2), −1)(2,\ 1638,\ (\ :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
7 1−T 1 - T
13 1+T 1 + T
good5 1−T+pT2 1 - T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+9T+pT2 1 + 9 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+11T+pT2 1 + 11 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+15T+pT2 1 + 15 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−11T+pT2 1 - 11 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−14T+pT2 1 - 14 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+14T+pT2 1 + 14 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

−9.067467943273849313007184265002, −7.935932481491448532694078819912, −7.80015052507622603521937526703, −6.69232029580076888718309416404, −5.63709318901757604410923700064, −5.14572588019451526117366164450, −3.74734603047492382743471605599, −2.56045395592179615052900667030, −1.71010590596796076771996481178, 0, 1.71010590596796076771996481178, 2.56045395592179615052900667030, 3.74734603047492382743471605599, 5.14572588019451526117366164450, 5.63709318901757604410923700064, 6.69232029580076888718309416404, 7.80015052507622603521937526703, 7.935932481491448532694078819912, 9.067467943273849313007184265002

Graph of the ZZ-function along the critical line