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Properties

Label 2-3192-1.1-c1-0-39
Degree 22
Conductor 31923192
Sign −1-1
Analytic cond. 25.488225.4882
Root an. cond. 5.048585.04858
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  − 3-s + 7-s + 9-s − 4·11-s + 2·13-s + 19-s − 21-s − 5·25-s − 27-s + 8·29-s − 8·31-s + 4·33-s − 10·37-s − 2·39-s + 2·41-s + 4·43-s − 2·47-s + 49-s + 12·53-s − 57-s − 4·59-s − 10·61-s + 63-s − 8·67-s + 6·71-s + 6·73-s + 5·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.229·19-s − 0.218·21-s − 25-s − 0.192·27-s + 1.48·29-s − 1.43·31-s + 0.696·33-s − 1.64·37-s − 0.320·39-s + 0.312·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.64·53-s − 0.132·57-s − 0.520·59-s − 1.28·61-s + 0.125·63-s − 0.977·67-s + 0.712·71-s + 0.702·73-s + 0.577·75-s + ⋯

Functional equation

Λ(s)=(3192s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3192s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31923192    =    23⋅3⋅7⋅192^{3} \cdot 3 \cdot 7 \cdot 19
Sign: −1-1
Analytic conductor: 25.488225.4882
Root analytic conductor: 5.048585.04858
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3192, ( :1/2), −1)(2,\ 3192,\ (\ :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 1
3 1+T 1 + T
7 1−T 1 - T
19 1−T 1 - T
good5 1+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+10T+pT2 1 + 10 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+10T+pT2 1 + 10 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 1−6T+pT2 1 - 6 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+6T+pT2 1 + 6 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

−8.227682736846371139757862218212, −7.54774603004303739744250845491, −6.83237077914798735999515558637, −5.82585188741248842436372335936, −5.36236997784956459297219291152, −4.52109255437283192936571124156, −3.59166116062660573192193622451, −2.50715584707627535123242612956, −1.41167507301882068370214918898, 0, 1.41167507301882068370214918898, 2.50715584707627535123242612956, 3.59166116062660573192193622451, 4.52109255437283192936571124156, 5.36236997784956459297219291152, 5.82585188741248842436372335936, 6.83237077914798735999515558637, 7.54774603004303739744250845491, 8.227682736846371139757862218212

Graph of the ZZ-function along the critical line