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Properties

Label 2-1638-1.1-c1-0-5
Degree 22
Conductor 16381638
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s + 3·11-s + 13-s − 14-s + 16-s + 2·19-s − 3·22-s + 3·23-s − 5·25-s − 26-s + 28-s + 5·31-s − 32-s − 7·37-s − 2·38-s − 3·41-s + 8·43-s + 3·44-s − 3·46-s + 3·47-s + 49-s + 5·50-s + 52-s + 12·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 0.639·22-s + 0.625·23-s − 25-s − 0.196·26-s + 0.188·28-s + 0.898·31-s − 0.176·32-s − 1.15·37-s − 0.324·38-s − 0.468·41-s + 1.21·43-s + 0.452·44-s − 0.442·46-s + 0.437·47-s + 1/7·49-s + 0.707·50-s + 0.138·52-s + 1.64·53-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: 11
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: 00
Selberg data: (2, 1638, ( :1/2), 1)(2,\ 1638,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3876537051.387653705
L(12)L(\frac12) ≈\approx 1.3876537051.387653705
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+pT2 1 + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1−5T+pT2 1 - 5 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+T+pT2 1 + T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−18T+pT2 1 - 18 T + p T^{2}
97 1−17T+pT2 1 - 17 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

−9.220725747608034123664354362313, −8.741104533844239670652837381474, −7.83867260492541076636187152090, −7.11996296423595540415557652746, −6.29215796679279225700527276455, −5.42538218373963254725865788850, −4.29341565444102448956917987575, −3.31582010907805915748552075894, −2.05606432181352469404790479184, −0.957671658928271127099414366745, 0.957671658928271127099414366745, 2.05606432181352469404790479184, 3.31582010907805915748552075894, 4.29341565444102448956917987575, 5.42538218373963254725865788850, 6.29215796679279225700527276455, 7.11996296423595540415557652746, 7.83867260492541076636187152090, 8.741104533844239670652837381474, 9.220725747608034123664354362313

Graph of the ZZ-function along the critical line