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Properties

Label 2-371280-1.1-c1-0-36
Degree 22
Conductor 371280371280
Sign 11
Analytic cond. 2964.682964.68
Root an. cond. 54.448954.4489
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 + 5-s + 7-s + 9-s + 4·11-s + 13-s − 15-s + 17-s − 4·19-s − 21-s − 8·23-s + 25-s − 27-s − 2·29-s − 4·33-s + 35-s + 6·37-s − 39-s + 10·41-s + 4·43-s + 45-s + 49-s − 51-s − 10·53-s + 4·55-s + 4·57-s + 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.140·51-s − 1.37·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + ⋯

Functional equation

Λ(s)=(371280s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(371280s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 371280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 371280371280    =    24⋅3⋅5⋅7⋅13⋅172^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17
Sign: 11
Analytic conductor: 2964.682964.68
Root analytic conductor: 54.448954.4489
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 371280, ( :1/2), 1)(2,\ 371280,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8139334622.813933462
L(12)L(\frac12) ≈\approx 2.8139334622.813933462
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
5 1−T 1 - T
7 1−T 1 - T
13 1−T 1 - T
17 1−T 1 - T
good11 1−4T+pT2 1 - 4 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−12.42536145394794, −12.02569769069637, −11.64223067919237, −11.08123196133471, −10.78684494490655, −10.32527605727014, −9.751988455861304, −9.261252076313470, −9.104806892082676, −8.349834745617292, −7.771905755524037, −7.647281732731432, −6.744947542183586, −6.332023521022173, −6.112892619298411, −5.649784049090289, −4.978191214446445, −4.468001954646831, −3.973011881428919, −3.698392045918998, −2.787556221245564, −2.120900318430339, −1.752610185417086, −1.066256697358743, −0.4921765066658509, 0.4921765066658509, 1.066256697358743, 1.752610185417086, 2.120900318430339, 2.787556221245564, 3.698392045918998, 3.973011881428919, 4.468001954646831, 4.978191214446445, 5.649784049090289, 6.112892619298411, 6.332023521022173, 6.744947542183586, 7.647281732731432, 7.771905755524037, 8.349834745617292, 9.104806892082676, 9.261252076313470, 9.751988455861304, 10.32527605727014, 10.78684494490655, 11.08123196133471, 11.64223067919237, 12.02569769069637, 12.42536145394794

Graph of the ZZ-function along the critical line