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Properties

Label 2-4160-1.1-c1-0-36
Degree 22
Conductor 41604160
Sign 11
Analytic cond. 33.217733.2177
Root an. cond. 5.763485.76348
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  + 2·3-s − 5-s + 9-s + 2·11-s − 13-s − 2·15-s + 2·17-s + 2·19-s − 2·23-s + 25-s − 4·27-s + 6·29-s − 2·31-s + 4·33-s + 6·37-s − 2·39-s + 2·41-s + 6·43-s − 45-s + 8·47-s − 7·49-s + 4·51-s + 2·53-s − 2·55-s + 4·57-s + 6·59-s + 14·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 0.277·13-s − 0.516·15-s + 0.485·17-s + 0.458·19-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.359·31-s + 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.312·41-s + 0.914·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.560·51-s + 0.274·53-s − 0.269·55-s + 0.529·57-s + 0.781·59-s + 1.79·61-s + ⋯

Functional equation

Λ(s)=(4160s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4160s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41604160    =    26⋅5⋅132^{6} \cdot 5 \cdot 13
Sign: 11
Analytic conductor: 33.217733.2177
Root analytic conductor: 5.763485.76348
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4160, ( :1/2), 1)(2,\ 4160,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7652960592.765296059
L(12)L(\frac12) ≈\approx 2.7652960592.765296059
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
5 1+T 1 + T
13 1+T 1 + T
good3 1−2T+pT2 1 - 2 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+10T+pT2 1 + 10 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 1−2T+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

−8.411115000826089753996710903934, −7.76319041513224922753287966748, −7.22602053082602449735325415510, −6.28727870819603758957301866812, −5.43479364046733402754835461064, −4.38796175153128331239805240266, −3.73594560911583979412114290925, −2.95784688889706959154847700703, −2.18201972761241363090254415749, −0.910977372178463170757222920731, 0.910977372178463170757222920731, 2.18201972761241363090254415749, 2.95784688889706959154847700703, 3.73594560911583979412114290925, 4.38796175153128331239805240266, 5.43479364046733402754835461064, 6.28727870819603758957301866812, 7.22602053082602449735325415510, 7.76319041513224922753287966748, 8.411115000826089753996710903934

Graph of the ZZ-function along the critical line