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Properties

Label 2-3332-1.1-c1-0-3
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 − 2·9-s − 5·11-s − 5·13-s − 17-s − 6·19-s + 4·23-s − 5·25-s + 5·27-s + 4·29-s + 5·33-s + 8·37-s + 5·39-s + 4·41-s − 6·43-s − 6·47-s + 51-s + 11·53-s + 6·57-s + 10·59-s + 10·67-s − 4·69-s + 9·71-s + 4·73-s + 5·75-s + 9·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 1.50·11-s − 1.38·13-s − 0.242·17-s − 1.37·19-s + 0.834·23-s − 25-s + 0.962·27-s + 0.742·29-s + 0.870·33-s + 1.31·37-s + 0.800·39-s + 0.624·41-s − 0.914·43-s − 0.875·47-s + 0.140·51-s + 1.51·53-s + 0.794·57-s + 1.30·59-s + 1.22·67-s − 0.481·69-s + 1.06·71-s + 0.468·73-s + 0.577·75-s + 1.01·79-s + 1/9·81-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 0.64030127440.6403012744
L(12)L(\frac12) ≈\approx 0.64030127440.6403012744
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+pT2 1 + p T^{2}
11 1+5T+pT2 1 + 5 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 1−4T+pT2 1 - 4 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1−9T+pT2 1 - 9 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−9T+pT2 1 - 9 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+15T+pT2 1 + 15 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

−8.374012440735736479868877176293, −8.042300812665714533613275999044, −7.04860562711275482218570288402, −6.38511506014092420612219539214, −5.40864521517896805635688508757, −5.04360908944665888730354415016, −4.11321335610142599569466903869, −2.73780970066283753303466245103, −2.31208754219170212371373460686, −0.45582208180966446071572557340, 0.45582208180966446071572557340, 2.31208754219170212371373460686, 2.73780970066283753303466245103, 4.11321335610142599569466903869, 5.04360908944665888730354415016, 5.40864521517896805635688508757, 6.38511506014092420612219539214, 7.04860562711275482218570288402, 8.042300812665714533613275999044, 8.374012440735736479868877176293

Graph of the ZZ-function along the critical line