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Properties

Label 2-320892-1.1-c1-0-1
Degree 22
Conductor 320892320892
Sign 11
Analytic cond. 2562.332562.33
Root an. cond. 50.619550.6195
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 − 4·7-s + 9-s − 13-s + 17-s + 2·19-s + 4·21-s − 5·25-s − 27-s + 2·29-s + 8·37-s + 39-s + 4·41-s − 4·43-s − 2·47-s + 9·49-s − 51-s − 6·53-s − 2·57-s − 6·59-s + 6·61-s − 4·63-s + 2·67-s + 5·75-s − 8·79-s + 81-s + 2·83-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.242·17-s + 0.458·19-s + 0.872·21-s − 25-s − 0.192·27-s + 0.371·29-s + 1.31·37-s + 0.160·39-s + 0.624·41-s − 0.609·43-s − 0.291·47-s + 9/7·49-s − 0.140·51-s − 0.824·53-s − 0.264·57-s − 0.781·59-s + 0.768·61-s − 0.503·63-s + 0.244·67-s + 0.577·75-s − 0.900·79-s + 1/9·81-s + 0.219·83-s + ⋯

Functional equation

Λ(s)=(320892s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(320892s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 320892320892    =    22⋅3⋅112⋅13⋅172^{2} \cdot 3 \cdot 11^{2} \cdot 13 \cdot 17
Sign: 11
Analytic conductor: 2562.332562.33
Root analytic conductor: 50.619550.6195
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 320892, ( :1/2), 1)(2,\ 320892,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.97162480130.9716248013
L(12)L(\frac12) ≈\approx 0.97162480130.9716248013
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
11 1 1
13 1+T 1 + T
17 1−T 1 - T
good5 1+pT2 1 + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−2T+pT2 1 - 2 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+4T+pT2 1 + 4 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+6T+pT2 1 + 6 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+pT2 1 + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−16T+pT2 1 - 16 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

−12.71457285630850, −12.03324147632738, −11.90697470390616, −11.24040163099333, −10.87667992089925, −10.17873209974291, −9.913385363497261, −9.519383271317379, −9.209324206995122, −8.462401651979755, −7.935537900478138, −7.442159097634523, −7.012114294218143, −6.399306093404742, −6.164373820456890, −5.663259832696918, −5.158769656424115, −4.487401388969493, −4.059765144465508, −3.401066205110604, −3.017966542909509, −2.422576856374139, −1.704443603996042, −0.9225664745386388, −0.3234111090829367, 0.3234111090829367, 0.9225664745386388, 1.704443603996042, 2.422576856374139, 3.017966542909509, 3.401066205110604, 4.059765144465508, 4.487401388969493, 5.158769656424115, 5.663259832696918, 6.164373820456890, 6.399306093404742, 7.012114294218143, 7.442159097634523, 7.935537900478138, 8.462401651979755, 9.209324206995122, 9.519383271317379, 9.913385363497261, 10.17873209974291, 10.87667992089925, 11.24040163099333, 11.90697470390616, 12.03324147632738, 12.71457285630850

Graph of the ZZ-function along the critical line