↑

Properties

Label 2-3332-68.67-c0-0-16
Degree 22
Conductor 33323332
Sign 11
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s + 11-s + 12-s − 13-s + 16-s + 17-s + 22-s − 2·23-s + 24-s + 25-s − 26-s − 27-s − 2·31-s + 32-s + 33-s + 34-s − 39-s + 44-s − 2·46-s + 48-s + 50-s + 51-s − 52-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s + 11-s + 12-s − 13-s + 16-s + 17-s + 22-s − 2·23-s + 24-s + 25-s − 26-s − 27-s − 2·31-s + 32-s + 33-s + 34-s − 39-s + 44-s − 2·46-s + 48-s + 50-s + 51-s − 52-s + ⋯

Functional equation

Λ(s)=(3332s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(3332s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    22⋅72⋅172^{2} \cdot 7^{2} \cdot 17
Sign: 11
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ3332(883,⋅)\chi_{3332} (883, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 1)(2,\ 3332,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 3.3112046803.311204680
L(12)L(\frac12) ≈\approx 3.3112046803.311204680
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−T 1 - T
7 1 1
17 1−T 1 - T
good3 1−T+T2 1 - T + T^{2}
5 (1−T)(1+T) ( 1 - T )( 1 + T )
11 1−T+T2 1 - T + T^{2}
13 1+T+T2 1 + T + T^{2}
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 (1+T)2 ( 1 + T )^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1+T)2 ( 1 + T )^{2}
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 (1−T)(1+T) ( 1 - T )( 1 + T )
53 1+T+T2 1 + T + T^{2}
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+T2 1 - T + T^{2}
83 (1−T)(1+T) ( 1 - T )( 1 + T )
89 1+T+T2 1 + T + T^{2}
97 (1−T)(1+T) ( 1 - T )( 1 + T )
show more
show less
   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

−8.750923436368834414603330497536, −7.83391507264668844340343737627, −7.43004858406289332920532983976, −6.48631318380755617525975688312, −5.72681217986316193188470379777, −4.92180405279032661637103741160, −3.88621534427499322473961829782, −3.44323487701105061188141418079, −2.45818357139276637353272527240, −1.67914071433243041950453158480, 1.67914071433243041950453158480, 2.45818357139276637353272527240, 3.44323487701105061188141418079, 3.88621534427499322473961829782, 4.92180405279032661637103741160, 5.72681217986316193188470379777, 6.48631318380755617525975688312, 7.43004858406289332920532983976, 7.83391507264668844340343737627, 8.750923436368834414603330497536

Graph of the ZZ-function along the critical line