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Properties

Label 2-684-19.18-c0-0-1
Degree 22
Conductor 684684
Sign 11
Analytic cond. 0.3413600.341360
Root an. cond. 0.5842600.584260
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  + 5-s − 7-s + 11-s + 17-s + 19-s − 2·23-s − 35-s − 43-s + 47-s + 55-s − 61-s − 73-s − 77-s − 2·83-s + 85-s + 95-s − 2·101-s − 2·115-s − 119-s + ⋯
L(s)  = 1  + 5-s − 7-s + 11-s + 17-s + 19-s − 2·23-s − 35-s − 43-s + 47-s + 55-s − 61-s − 73-s − 77-s − 2·83-s + 85-s + 95-s − 2·101-s − 2·115-s − 119-s + ⋯

Functional equation

Λ(s)=(684s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(684s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 684684    =    22⋅32⋅192^{2} \cdot 3^{2} \cdot 19
Sign: 11
Analytic conductor: 0.3413600.341360
Root analytic conductor: 0.5842600.584260
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ684(37,⋅)\chi_{684} (37, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 684, ( :0), 1)(2,\ 684,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0498061511.049806151
L(12)L(\frac12) ≈\approx 1.0498061511.049806151
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
3 1 1
19 1−T 1 - T
good5 1−T+T2 1 - T + T^{2}
7 1+T+T2 1 + T + T^{2}
11 1−T+T2 1 - T + T^{2}
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 1−T+T2 1 - T + T^{2}
23 (1+T)2 ( 1 + T )^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
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+T2 1 + T + T^{2}
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 1+T+T2 1 + T + T^{2}
79 (1−T)(1+T) ( 1 - T )( 1 + T )
83 (1+T)2 ( 1 + T )^{2}
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

−10.35491111494555031411626029725, −9.758818438700118784693109349550, −9.314387812062134939433397223502, −8.117394606784884318841001905600, −7.03087957755923141913663523242, −6.10749108764499485087312327875, −5.59033728809514506575945817746, −4.06732770039287226268176886749, −3.05201959040369072688091579245, −1.62495726525899861887525643650, 1.62495726525899861887525643650, 3.05201959040369072688091579245, 4.06732770039287226268176886749, 5.59033728809514506575945817746, 6.10749108764499485087312327875, 7.03087957755923141913663523242, 8.117394606784884318841001905600, 9.314387812062134939433397223502, 9.758818438700118784693109349550, 10.35491111494555031411626029725

Graph of the ZZ-function along the critical line