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Properties

Label 2-2700-3.2-c0-0-0
Degree 22
Conductor 27002700
Sign 11
Analytic cond. 1.347471.34747
Root an. cond. 1.160801.16080
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  − 2·7-s + 13-s + 2·19-s − 31-s + 37-s + 43-s + 3·49-s + 2·61-s + 67-s + 73-s − 79-s − 2·91-s − 2·97-s + 103-s − 109-s + ⋯
L(s)  = 1  − 2·7-s + 13-s + 2·19-s − 31-s + 37-s + 43-s + 3·49-s + 2·61-s + 67-s + 73-s − 79-s − 2·91-s − 2·97-s + 103-s − 109-s + ⋯

Functional equation

Λ(s)=(2700s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(2700s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27002700    =    22⋅33⋅522^{2} \cdot 3^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 1.347471.34747
Root analytic conductor: 1.160801.16080
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ2700(701,⋅)\chi_{2700} (701, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2700, ( :0), 1)(2,\ 2700,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0249872311.024987231
L(12)L(\frac12) ≈\approx 1.0249872311.024987231
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
5 1 1
good7 (1+T)2 ( 1 + T )^{2}
11 (1−T)(1+T) ( 1 - T )( 1 + T )
13 1−T+T2 1 - T + T^{2}
17 (1−T)(1+T) ( 1 - T )( 1 + T )
19 (1−T)2 ( 1 - T )^{2}
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 1+T+T2 1 + T + 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)(1+T) ( 1 - T )( 1 + T )
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 (1−T)(1+T) ( 1 - T )( 1 + T )
61 (1−T)2 ( 1 - T )^{2}
67 1−T+T2 1 - T + T^{2}
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 1−T+T2 1 - T + T^{2}
79 1+T+T2 1 + T + T^{2}
83 (1−T)(1+T) ( 1 - T )( 1 + T )
89 (1−T)(1+T) ( 1 - T )( 1 + T )
97 (1+T)2 ( 1 + 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

−9.369894900461193934825877715407, −8.315847245919080541688146615858, −7.33552088663502316587376717662, −6.76350900263612993765666679587, −5.93419497994444199209392128140, −5.40533598838664707644239047412, −3.95439187298776558510360491013, −3.41610161486337847990248745465, −2.59911500477229605674785362844, −0.947520220347311601026913039911, 0.947520220347311601026913039911, 2.59911500477229605674785362844, 3.41610161486337847990248745465, 3.95439187298776558510360491013, 5.40533598838664707644239047412, 5.93419497994444199209392128140, 6.76350900263612993765666679587, 7.33552088663502316587376717662, 8.315847245919080541688146615858, 9.369894900461193934825877715407

Graph of the ZZ-function along the critical line