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Properties

Label 2-416-104.51-c0-0-1
Degree 22
Conductor 416416
Sign 11
Analytic cond. 0.2076110.207611
Root an. cond. 0.4556430.455643
Motivic weight 00
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 − 13-s + 15-s − 17-s − 21-s − 27-s + 2·31-s − 35-s + 37-s − 39-s + 43-s − 47-s − 51-s − 65-s − 71-s − 81-s − 85-s + 91-s + 2·93-s − 105-s − 2·107-s + 109-s + 111-s + 2·113-s + 119-s + ⋯
L(s)  = 1  + 3-s + 5-s − 7-s − 13-s + 15-s − 17-s − 21-s − 27-s + 2·31-s − 35-s + 37-s − 39-s + 43-s − 47-s − 51-s − 65-s − 71-s − 81-s − 85-s + 91-s + 2·93-s − 105-s − 2·107-s + 109-s + 111-s + 2·113-s + 119-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: 11
Analytic conductor: 0.2076110.207611
Root analytic conductor: 0.4556430.455643
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ416(207,⋅)\chi_{416} (207, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 416, ( :0), 1)(2,\ 416,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1060538651.106053865
L(12)L(\frac12) ≈\approx 1.1060538651.106053865
L(1)L(1) 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
13 1+T 1 + T
good3 1−T+T2 1 - T + T^{2}
5 1−T+T2 1 - T + T^{2}
7 1+T+T2 1 + T + T^{2}
11 (1−T)(1+T) ( 1 - T )( 1 + T )
17 1+T+T2 1 + T + T^{2}
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1−T)2 ( 1 - T )^{2}
37 1−T+T2 1 - T + T^{2}
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 1−T+T2 1 - T + T^{2}
47 1+T+T2 1 + T + T^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 (1−T)(1+T) ( 1 - T )( 1 + T )
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 1+T+T2 1 + T + T^{2}
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 (1−T)(1+T) ( 1 - T )( 1 + T )
83 (1−T)(1+T) ( 1 - T )( 1 + T )
89 (1−T)(1+T) ( 1 - T )( 1 + T )
97 (1−T)(1+T) ( 1 - T )( 1 + T )
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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

−11.42371445012146906223298386792, −10.10071388774337285274400321869, −9.624135009704371750161913168064, −8.889879139031364702347720607118, −7.86411890344124420829049809431, −6.69492788052190547853235166609, −5.88206968753216184950267243811, −4.48916427945045909880908525906, −3.01462645230284483610556303432, −2.26480681637905692621118531044, 2.26480681637905692621118531044, 3.01462645230284483610556303432, 4.48916427945045909880908525906, 5.88206968753216184950267243811, 6.69492788052190547853235166609, 7.86411890344124420829049809431, 8.889879139031364702347720607118, 9.624135009704371750161913168064, 10.10071388774337285274400321869, 11.42371445012146906223298386792

Graph of the ZZ-function along the critical line