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Properties

Label 2-3360-1.1-c1-0-10
Degree 22
Conductor 33603360
Sign 11
Analytic cond. 26.829726.8297
Root an. cond. 5.179745.17974
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 − 2·13-s − 15-s − 2·17-s + 8·19-s + 21-s + 25-s − 27-s − 2·29-s − 4·33-s − 35-s − 6·37-s + 2·39-s + 2·41-s + 4·43-s + 45-s + 8·47-s + 49-s + 2·51-s − 6·53-s + 4·55-s − 8·57-s − 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s + 1.83·19-s + 0.218·21-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 + 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.539·55-s − 1.05·57-s − 0.256·61-s + ⋯

Functional equation

Λ(s)=(3360s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3360s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33603360    =    25⋅3⋅5⋅72^{5} \cdot 3 \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 26.829726.8297
Root analytic conductor: 5.179745.17974
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3360, ( :1/2), 1)(2,\ 3360,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6825987141.682598714
L(12)L(\frac12) ≈\approx 1.6825987141.682598714
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
good11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−8T+pT2 1 - 8 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+6T+pT2 1 + 6 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−18T+pT2 1 - 18 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−8.890905071405940263807660308258, −7.64478753337256159133360922574, −7.06086036239019688063680452315, −6.35316640417836655320198380518, −5.63632782549750996248123136110, −4.91380017540757573217630253175, −3.97110146144033917413647839403, −3.09040679008008900975048055760, −1.89425390748404433456997372556, −0.818727734574118390759677993169, 0.818727734574118390759677993169, 1.89425390748404433456997372556, 3.09040679008008900975048055760, 3.97110146144033917413647839403, 4.91380017540757573217630253175, 5.63632782549750996248123136110, 6.35316640417836655320198380518, 7.06086036239019688063680452315, 7.64478753337256159133360922574, 8.890905071405940263807660308258

Graph of the ZZ-function along the critical line