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Properties

Label 2-3360-1.1-c1-0-9
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 + 4·19-s + 21-s + 25-s + 27-s + 10·29-s − 4·33-s − 35-s + 6·37-s − 2·39-s − 10·41-s + 8·43-s − 45-s + 4·47-s + 49-s − 2·51-s + 6·53-s + 4·55-s + 4·57-s + 10·61-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 + 0.917·19-s + 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.696·33-s − 0.169·35-s + 0.986·37-s − 0.320·39-s − 1.56·41-s + 1.21·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + 1.28·61-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 2.0010494872.001049487
L(12)L(\frac12) ≈\approx 2.0010494872.001049487
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 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1−10T+pT2 1 - 10 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 1−8T+pT2 1 - 8 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1−8T+pT2 1 - 8 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 1−6T+pT2 1 - 6 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

−8.437871548224310202742352185353, −7.949152959733991878916874270549, −7.31979595951381878058034312516, −6.55341822008387752471814319456, −5.36060461079758519453871924476, −4.81491928124302049005015756014, −3.92742709962574332759519972128, −2.88633232471614386188270893470, −2.28839099576648481139527420277, −0.813170101248483559516930675379, 0.813170101248483559516930675379, 2.28839099576648481139527420277, 2.88633232471614386188270893470, 3.92742709962574332759519972128, 4.81491928124302049005015756014, 5.36060461079758519453871924476, 6.55341822008387752471814319456, 7.31979595951381878058034312516, 7.949152959733991878916874270549, 8.437871548224310202742352185353

Graph of the ZZ-function along the critical line