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Properties

Label 2-320892-1.1-c1-0-13
Degree 22
Conductor 320892320892
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 9-s + 13-s − 2·15-s − 17-s + 2·19-s − 25-s + 27-s − 2·29-s + 8·31-s + 4·37-s + 39-s − 10·41-s − 2·45-s − 7·49-s − 51-s + 6·53-s + 2·57-s − 14·61-s − 2·65-s − 6·67-s + 6·71-s − 4·73-s − 75-s + 8·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.277·13-s − 0.516·15-s − 0.242·17-s + 0.458·19-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.657·37-s + 0.160·39-s − 1.56·41-s − 0.298·45-s − 49-s − 0.140·51-s + 0.824·53-s + 0.264·57-s − 1.79·61-s − 0.248·65-s − 0.733·67-s + 0.712·71-s − 0.468·73-s − 0.115·75-s + 0.900·79-s + 1/9·81-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: −1-1
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: 11
Selberg data: (2, 320892, ( :1/2), −1)(2,\ 320892,\ (\ :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
11 1 1
13 1−T 1 - T
17 1+T 1 + T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+pT2 1 + 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 1−8T+pT2 1 - 8 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1+6T+pT2 1 + 6 T + p T^{2}
71 1−6T+pT2 1 - 6 T + 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+4T+pT2 1 + 4 T + p T^{2}
89 1+12T+pT2 1 + 12 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

−12.88440717315032, −12.32031663970101, −11.89649074813707, −11.60668360373167, −11.05409497980581, −10.64000386329240, −9.971138651902090, −9.732863222110321, −9.142170849995676, −8.554931198149262, −8.331383346987121, −7.713146670441039, −7.526888342107864, −6.806850020798795, −6.459856562899966, −5.844369741526736, −5.233654591682197, −4.636688415225073, −4.237833877879181, −3.757665586887557, −3.093734599666578, −2.896965627817661, −2.009578493822262, −1.492050337943157, −0.7449930965738251, 0, 0.7449930965738251, 1.492050337943157, 2.009578493822262, 2.896965627817661, 3.093734599666578, 3.757665586887557, 4.237833877879181, 4.636688415225073, 5.233654591682197, 5.844369741526736, 6.459856562899966, 6.806850020798795, 7.526888342107864, 7.713146670441039, 8.331383346987121, 8.554931198149262, 9.142170849995676, 9.732863222110321, 9.971138651902090, 10.64000386329240, 11.05409497980581, 11.60668360373167, 11.89649074813707, 12.32031663970101, 12.88440717315032

Graph of the ZZ-function along the critical line