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Properties

Label 2-588-1.1-c1-0-2
Degree 22
Conductor 588588
Sign 11
Analytic cond. 4.695204.69520
Root an. cond. 2.166842.16684
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 + 2·5-s + 9-s + 2·11-s − 4·13-s + 2·15-s + 6·17-s + 8·19-s − 6·23-s − 25-s + 27-s − 10·29-s + 4·31-s + 2·33-s + 6·37-s − 4·39-s − 6·41-s + 4·43-s + 2·45-s + 8·47-s + 6·51-s + 2·53-s + 4·55-s + 8·57-s − 4·59-s − 8·61-s − 8·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.516·15-s + 1.45·17-s + 1.83·19-s − 1.25·23-s − 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.348·33-s + 0.986·37-s − 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 0.840·51-s + 0.274·53-s + 0.539·55-s + 1.05·57-s − 0.520·59-s − 1.02·61-s − 0.992·65-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 11
Analytic conductor: 4.695204.69520
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 588, ( :1/2), 1)(2,\ 588,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1088546132.108854613
L(12)L(\frac12) ≈\approx 2.1088546132.108854613
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 1
good5 1−2T+pT2 1 - 2 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−4T+pT2 1 - 4 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

−10.36611036627762852540629826196, −9.583707885022017939372743886143, −9.362116383379991779782320758654, −7.85267446574644619430762393574, −7.37885248339416856965042328499, −6.01050196685275589688564643658, −5.28904041113039493355326669628, −3.90419895713753879761270516632, −2.76100450230404433527159597103, −1.51338087290757062643114219385, 1.51338087290757062643114219385, 2.76100450230404433527159597103, 3.90419895713753879761270516632, 5.28904041113039493355326669628, 6.01050196685275589688564643658, 7.37885248339416856965042328499, 7.85267446574644619430762393574, 9.362116383379991779782320758654, 9.583707885022017939372743886143, 10.36611036627762852540629826196

Graph of the ZZ-function along the critical line