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Properties

Label 2-1984-31.30-c0-0-0
Degree 22
Conductor 19841984
Sign 11
Analytic cond. 0.9901440.990144
Root an. cond. 0.9950600.995060
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 + 9-s + 19-s + 31-s − 35-s − 41-s + 45-s + 2·47-s + 59-s − 63-s − 2·67-s − 71-s + 81-s + 95-s − 97-s + 101-s − 103-s + 107-s + 109-s − 113-s + ⋯
L(s)  = 1  + 5-s − 7-s + 9-s + 19-s + 31-s − 35-s − 41-s + 45-s + 2·47-s + 59-s − 63-s − 2·67-s − 71-s + 81-s + 95-s − 97-s + 101-s − 103-s + 107-s + 109-s − 113-s + ⋯

Functional equation

Λ(s)=(1984s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1984s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19841984    =    26⋅312^{6} \cdot 31
Sign: 11
Analytic conductor: 0.9901440.990144
Root analytic conductor: 0.9950600.995060
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ1984(1921,⋅)\chi_{1984} (1921, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1984, ( :0), 1)(2,\ 1984,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3639706091.363970609
L(12)L(\frac12) ≈\approx 1.3639706091.363970609
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
31 1−T 1 - T
good3 (1−T)(1+T) ( 1 - T )( 1 + T )
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 )
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 (1−T)(1+T) ( 1 - T )( 1 + T )
19 1−T+T2 1 - T + T^{2}
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 1+T+T2 1 + T + T^{2}
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 (1−T)2 ( 1 - T )^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 1−T+T2 1 - T + T^{2}
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 (1+T)2 ( 1 + T )^{2}
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+T2 1 + T + 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

−9.498562370271284513168455392276, −8.809220719804173880707610068874, −7.63587780772854693400670068344, −6.93801372505915564084401401634, −6.20032309942401690853112582952, −5.49393077538779529026821463484, −4.47888650250490094258310220839, −3.45949718743552220459799941634, −2.47767351245492657633422028026, −1.29804537939601829628431090955, 1.29804537939601829628431090955, 2.47767351245492657633422028026, 3.45949718743552220459799941634, 4.47888650250490094258310220839, 5.49393077538779529026821463484, 6.20032309942401690853112582952, 6.93801372505915564084401401634, 7.63587780772854693400670068344, 8.809220719804173880707610068874, 9.498562370271284513168455392276

Graph of the ZZ-function along the critical line