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Properties

Label 2-3332-1.1-c1-0-5
Degree 22
Conductor 33323332
Sign 11
Analytic cond. 26.606126.6061
Root an. cond. 5.158115.15811
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  + 3-s − 4·5-s − 2·9-s − 3·11-s − 5·13-s − 4·15-s − 17-s + 6·19-s − 4·23-s + 11·25-s − 5·27-s + 8·29-s − 3·33-s − 8·37-s − 5·39-s − 8·41-s + 10·43-s + 8·45-s − 2·47-s − 51-s + 3·53-s + 12·55-s + 6·57-s + 2·59-s − 8·61-s + 20·65-s + 14·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s − 2/3·9-s − 0.904·11-s − 1.38·13-s − 1.03·15-s − 0.242·17-s + 1.37·19-s − 0.834·23-s + 11/5·25-s − 0.962·27-s + 1.48·29-s − 0.522·33-s − 1.31·37-s − 0.800·39-s − 1.24·41-s + 1.52·43-s + 1.19·45-s − 0.291·47-s − 0.140·51-s + 0.412·53-s + 1.61·55-s + 0.794·57-s + 0.260·59-s − 1.02·61-s + 2.48·65-s + 1.71·67-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 11
Analytic conductor: 26.606126.6061
Root analytic conductor: 5.158115.15811
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3332, ( :1/2), 1)(2,\ 3332,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.80627546360.8062754636
L(12)L(\frac12) ≈\approx 0.80627546360.8062754636
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
7 1 1
17 1+T 1 + T
good3 1−T+pT2 1 - T + p T^{2}
5 1+4T+pT2 1 + 4 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+5T+pT2 1 + 5 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1−3T+pT2 1 - 3 T + p T^{2}
59 1−2T+pT2 1 - 2 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1−7T+pT2 1 - 7 T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1−15T+pT2 1 - 15 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−T+pT2 1 - T + p T^{2}
97 1−18T+pT2 1 - 18 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

−8.412207524319629505390518493394, −7.83914240649366228838486958135, −7.49313014608019165765050404536, −6.64060448648621865259785379284, −5.29448629237218268670726784084, −4.81108611409071186497772433095, −3.76859904237221605053167419340, −3.10298042904571192545973628388, −2.36237586664771651599690165474, −0.48857560730778819962298282627, 0.48857560730778819962298282627, 2.36237586664771651599690165474, 3.10298042904571192545973628388, 3.76859904237221605053167419340, 4.81108611409071186497772433095, 5.29448629237218268670726784084, 6.64060448648621865259785379284, 7.49313014608019165765050404536, 7.83914240649366228838486958135, 8.412207524319629505390518493394

Graph of the ZZ-function along the critical line