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Properties

Label 2-1638-1.1-c1-0-27
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 − 3·5-s + 7-s + 8-s − 3·10-s − 11-s − 13-s + 14-s + 16-s − 5·17-s − 19-s − 3·20-s − 22-s + 23-s + 4·25-s − 26-s + 28-s − 7·29-s − 2·31-s + 32-s − 5·34-s − 3·35-s + 37-s − 38-s − 3·40-s − 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.229·19-s − 0.670·20-s − 0.213·22-s + 0.208·23-s + 4/5·25-s − 0.196·26-s + 0.188·28-s − 1.29·29-s − 0.359·31-s + 0.176·32-s − 0.857·34-s − 0.507·35-s + 0.164·37-s − 0.162·38-s − 0.474·40-s − 0.937·41-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+3T+pT2 1 + 3 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
17 1+5T+pT2 1 + 5 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+7T+pT2 1 + 7 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+3T+pT2 1 + 3 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 1−6T+pT2 1 - 6 T + p T^{2}
61 1+5T+pT2 1 + 5 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1+14T+pT2 1 + 14 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1+10T+pT2 1 + 10 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

−8.753348500037287827964908129272, −8.087399671123924969643801147843, −7.31743211521265601126957812633, −6.69061749873854041708189827769, −5.51931339869349074764387757967, −4.66162486191056383154167071598, −4.01680586816339694050615224274, −3.11346507231815676323880972931, −1.89913794199562214175475004949, 0, 1.89913794199562214175475004949, 3.11346507231815676323880972931, 4.01680586816339694050615224274, 4.66162486191056383154167071598, 5.51931339869349074764387757967, 6.69061749873854041708189827769, 7.31743211521265601126957812633, 8.087399671123924969643801147843, 8.753348500037287827964908129272

Graph of the ZZ-function along the critical line