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Properties

Label 2-1638-1.1-c1-0-8
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 + 4·5-s − 7-s − 8-s − 4·10-s − 11-s − 13-s + 14-s + 16-s − 6·19-s + 4·20-s + 22-s + 7·23-s + 11·25-s + 26-s − 28-s + 4·29-s + 7·31-s − 32-s − 4·35-s + 9·37-s + 6·38-s − 4·40-s + 3·41-s + 4·43-s − 44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.377·7-s − 0.353·8-s − 1.26·10-s − 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.37·19-s + 0.894·20-s + 0.213·22-s + 1.45·23-s + 11/5·25-s + 0.196·26-s − 0.188·28-s + 0.742·29-s + 1.25·31-s − 0.176·32-s − 0.676·35-s + 1.47·37-s + 0.973·38-s − 0.632·40-s + 0.468·41-s + 0.609·43-s − 0.150·44-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 1.6502334641.650233464
L(12)L(\frac12) ≈\approx 1.6502334641.650233464
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 1−4T+pT2 1 - 4 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1−7T+pT2 1 - 7 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1−9T+pT2 1 - 9 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+7T+pT2 1 + 7 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1−T+pT2 1 - T + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1−5T+pT2 1 - 5 T + p T^{2}
79 1−11T+pT2 1 - 11 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+T+pT2 1 + 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

−9.382683634490519654764199003796, −8.842134311662064724454282147488, −7.943345217051475117643242740083, −6.72942118020662831970006205526, −6.37840271109207786796552495603, −5.49386334296658824475642157597, −4.55256625494296783763579353487, −2.85733179130965228288861874054, −2.28863422577859859560688974167, −1.03297262712619069700335338528, 1.03297262712619069700335338528, 2.28863422577859859560688974167, 2.85733179130965228288861874054, 4.55256625494296783763579353487, 5.49386334296658824475642157597, 6.37840271109207786796552495603, 6.72942118020662831970006205526, 7.943345217051475117643242740083, 8.842134311662064724454282147488, 9.382683634490519654764199003796

Graph of the ZZ-function along the critical line