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Properties

Label 2-3332-1.1-c1-0-21
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  − 3·3-s + 4·5-s + 6·9-s + 11-s + 3·13-s − 12·15-s − 17-s − 2·19-s + 4·23-s + 11·25-s − 9·27-s + 8·31-s − 3·33-s + 8·37-s − 9·39-s + 10·43-s + 24·45-s − 10·47-s + 3·51-s + 3·53-s + 4·55-s + 6·57-s − 14·59-s − 8·61-s + 12·65-s − 10·67-s − 12·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.78·5-s + 2·9-s + 0.301·11-s + 0.832·13-s − 3.09·15-s − 0.242·17-s − 0.458·19-s + 0.834·23-s + 11/5·25-s − 1.73·27-s + 1.43·31-s − 0.522·33-s + 1.31·37-s − 1.44·39-s + 1.52·43-s + 3.57·45-s − 1.45·47-s + 0.420·51-s + 0.412·53-s + 0.539·55-s + 0.794·57-s − 1.82·59-s − 1.02·61-s + 1.48·65-s − 1.22·67-s − 1.44·69-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 1.6638200841.663820084
L(12)L(\frac12) ≈\approx 1.6638200841.663820084
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+pT+pT2 1 + p T + p T^{2}
5 1−4T+pT2 1 - 4 T + p T^{2}
11 1−T+pT2 1 - T + p T^{2}
13 1−3T+pT2 1 - 3 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+10T+pT2 1 + 10 T + p T^{2}
53 1−3T+pT2 1 - 3 T + p T^{2}
59 1+14T+pT2 1 + 14 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+5T+pT2 1 + 5 T + p T^{2}
73 1+16T+pT2 1 + 16 T + p T^{2}
79 1−11T+pT2 1 - 11 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−9T+pT2 1 - 9 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

−8.973936402482286354207399587458, −7.69024035678898130149858331901, −6.60165596308849873907420846454, −6.22078735832051372202585427283, −5.87479884156858091839879278168, −4.93772697182014911636657086789, −4.40820492715886183282953413943, −2.87859951484990752955258598597, −1.67453831099288359567119266994, −0.914365510636910812510763279545, 0.914365510636910812510763279545, 1.67453831099288359567119266994, 2.87859951484990752955258598597, 4.40820492715886183282953413943, 4.93772697182014911636657086789, 5.87479884156858091839879278168, 6.22078735832051372202585427283, 6.60165596308849873907420846454, 7.69024035678898130149858331901, 8.973936402482286354207399587458

Graph of the ZZ-function along the critical line