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Properties

Label 2-405-1.1-c1-0-10
Degree 22
Conductor 405405
Sign −1-1
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
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  − 2-s − 4-s + 5-s − 3·7-s + 3·8-s − 10-s + 2·11-s − 2·13-s + 3·14-s − 16-s − 4·17-s − 8·19-s − 20-s − 2·22-s − 3·23-s + 25-s + 2·26-s + 3·28-s + 29-s − 5·32-s + 4·34-s − 3·35-s − 4·37-s + 8·38-s + 3·40-s − 5·41-s − 8·43-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.13·7-s + 1.06·8-s − 0.316·10-s + 0.603·11-s − 0.554·13-s + 0.801·14-s − 1/4·16-s − 0.970·17-s − 1.83·19-s − 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s + 0.392·26-s + 0.566·28-s + 0.185·29-s − 0.883·32-s + 0.685·34-s − 0.507·35-s − 0.657·37-s + 1.29·38-s + 0.474·40-s − 0.780·41-s − 1.21·43-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: −1-1
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 405, ( :1/2), −1)(2,\ 405,\ (\ :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)
bad3 1 1
5 1−T 1 - T
good2 1+T+pT2 1 + T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1−T+pT2 1 - T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1+5T+pT2 1 + 5 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+7T+pT2 1 + 7 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−14T+pT2 1 - 14 T + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1+3T+pT2 1 + 3 T + p T^{2}
71 1+2T+pT2 1 + 2 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+6T+pT2 1 + 6 T + p T^{2}
83 1+9T+pT2 1 + 9 T + p T^{2}
89 1−15T+pT2 1 - 15 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

−10.32043257877866718022672062677, −9.946177052534403132870451803520, −8.964305577132556078601253128051, −8.406907006225460062267410041104, −6.95907981187563203473208917697, −6.29557968587596560657178728455, −4.84628635731234228377894729907, −3.75836640301739809806939141128, −2.06014522909482997436709887093, 0, 2.06014522909482997436709887093, 3.75836640301739809806939141128, 4.84628635731234228377894729907, 6.29557968587596560657178728455, 6.95907981187563203473208917697, 8.406907006225460062267410041104, 8.964305577132556078601253128051, 9.946177052534403132870451803520, 10.32043257877866718022672062677

Graph of the ZZ-function along the critical line