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Properties

Label 2-5800-1.1-c1-0-51
Degree 22
Conductor 58005800
Sign −1-1
Analytic cond. 46.313246.3132
Root an. cond. 6.805386.80538
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  − 2·3-s − 4·7-s + 9-s − 6·13-s + 8·21-s + 8·23-s + 4·27-s − 29-s + 8·31-s + 12·39-s + 6·41-s + 2·43-s + 2·47-s + 9·49-s + 6·53-s + 12·59-s − 6·61-s − 4·63-s − 16·67-s − 16·69-s − 4·73-s − 8·79-s − 11·81-s − 8·83-s + 2·87-s − 10·89-s + 24·91-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.51·7-s + 1/3·9-s − 1.66·13-s + 1.74·21-s + 1.66·23-s + 0.769·27-s − 0.185·29-s + 1.43·31-s + 1.92·39-s + 0.937·41-s + 0.304·43-s + 0.291·47-s + 9/7·49-s + 0.824·53-s + 1.56·59-s − 0.768·61-s − 0.503·63-s − 1.95·67-s − 1.92·69-s − 0.468·73-s − 0.900·79-s − 1.22·81-s − 0.878·83-s + 0.214·87-s − 1.05·89-s + 2.51·91-s + ⋯

Functional equation

Λ(s)=(5800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 58005800    =    23⋅52⋅292^{3} \cdot 5^{2} \cdot 29
Sign: −1-1
Analytic conductor: 46.313246.3132
Root analytic conductor: 6.805386.80538
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5800, ( :1/2), −1)(2,\ 5800,\ (\ :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
5 1 1
29 1+T 1 + T
good3 1+2T+pT2 1 + 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1+pT2 1 + p T^{2}
41 1−6T+pT2 1 - 6 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 1−6T+pT2 1 - 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+16T+pT2 1 + 16 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−7.31017073685675572016217862806, −7.06858877722135191127773358539, −6.25567243109627997196809723557, −5.70726895584028784978653649653, −4.94368401696947777983396039060, −4.28690715149454511878958416902, −3.04124127274992298758499240166, −2.59669004146168747577806606099, −0.906127559401400389456999505674, 0, 0.906127559401400389456999505674, 2.59669004146168747577806606099, 3.04124127274992298758499240166, 4.28690715149454511878958416902, 4.94368401696947777983396039060, 5.70726895584028784978653649653, 6.25567243109627997196809723557, 7.06858877722135191127773358539, 7.31017073685675572016217862806

Graph of the ZZ-function along the critical line