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Properties

Label 2-4704-1.1-c1-0-61
Degree 22
Conductor 47044704
Sign −1-1
Analytic cond. 37.561637.5616
Root an. cond. 6.128756.12875
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 − 3·5-s + 9-s + 11-s + 4·13-s − 3·15-s − 4·17-s − 8·23-s + 4·25-s + 27-s − 7·29-s + 11·31-s + 33-s + 4·37-s + 4·39-s + 4·41-s + 2·43-s − 3·45-s − 2·47-s − 4·51-s − 11·53-s − 3·55-s + 7·59-s − 10·61-s − 12·65-s − 10·67-s − 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.774·15-s − 0.970·17-s − 1.66·23-s + 4/5·25-s + 0.192·27-s − 1.29·29-s + 1.97·31-s + 0.174·33-s + 0.657·37-s + 0.640·39-s + 0.624·41-s + 0.304·43-s − 0.447·45-s − 0.291·47-s − 0.560·51-s − 1.51·53-s − 0.404·55-s + 0.911·59-s − 1.28·61-s − 1.48·65-s − 1.22·67-s − 0.963·69-s + ⋯

Functional equation

Λ(s)=(4704s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4704s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47044704    =    25⋅3⋅722^{5} \cdot 3 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 37.561637.5616
Root analytic conductor: 6.128756.12875
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4704, ( :1/2), −1)(2,\ 4704,\ (\ :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 1
good5 1+3T+pT2 1 + 3 T + p T^{2}
11 1−T+pT2 1 - T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+7T+pT2 1 + 7 T + p T^{2}
31 1−11T+pT2 1 - 11 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1+11T+pT2 1 + 11 T + p T^{2}
59 1−7T+pT2 1 - 7 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+10T+pT2 1 + 10 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+11T+pT2 1 + 11 T + p T^{2}
83 1−11T+pT2 1 - 11 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+7T+pT2 1 + 7 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

−7.984331877365611098322205043752, −7.47550962443046423122867307670, −6.48923992128165791411727457072, −5.96251327588168626685822102926, −4.57469865580799387365402434132, −4.11507132647141149122813982159, −3.50490904927474427804969626521, −2.53078320913080985896661665550, −1.37769040801800048545486958028, 0, 1.37769040801800048545486958028, 2.53078320913080985896661665550, 3.50490904927474427804969626521, 4.11507132647141149122813982159, 4.57469865580799387365402434132, 5.96251327588168626685822102926, 6.48923992128165791411727457072, 7.47550962443046423122867307670, 7.984331877365611098322205043752

Graph of the ZZ-function along the critical line