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Properties

Label 2-1512-1.1-c1-0-4
Degree 22
Conductor 15121512
Sign 11
Analytic cond. 12.073312.0733
Root an. cond. 3.474673.47467
Motivic weight 11
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 − 4·13-s + 6·17-s − 19-s + 9·23-s − 4·25-s + 11·31-s + 35-s + 37-s + 5·41-s + 2·43-s − 2·47-s + 49-s + 12·53-s + 55-s − 6·59-s + 4·65-s − 4·67-s + 13·71-s − 14·73-s + 77-s + 10·79-s + 8·83-s − 6·85-s − 3·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.301·11-s − 1.10·13-s + 1.45·17-s − 0.229·19-s + 1.87·23-s − 4/5·25-s + 1.97·31-s + 0.169·35-s + 0.164·37-s + 0.780·41-s + 0.304·43-s − 0.291·47-s + 1/7·49-s + 1.64·53-s + 0.134·55-s − 0.781·59-s + 0.496·65-s − 0.488·67-s + 1.54·71-s − 1.63·73-s + 0.113·77-s + 1.12·79-s + 0.878·83-s − 0.650·85-s − 0.317·89-s + ⋯

Functional equation

Λ(s)=(1512s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1512s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15121512    =    23⋅33⋅72^{3} \cdot 3^{3} \cdot 7
Sign: 11
Analytic conductor: 12.073312.0733
Root analytic conductor: 3.474673.47467
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1512, ( :1/2), 1)(2,\ 1512,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3843461081.384346108
L(12)L(\frac12) ≈\approx 1.3843461081.384346108
L(32)L(\frac{3}{2}) 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
7 1+T 1 + T
good5 1+T+pT2 1 + T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−9T+pT2 1 - 9 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−11T+pT2 1 - 11 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+2T+pT2 1 + 2 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−13T+pT2 1 - 13 T + p T^{2}
73 1+14T+pT2 1 + 14 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1+3T+pT2 1 + 3 T + p T^{2}
97 1+pT2 1 + p 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

−9.608203759312194453797257789404, −8.645837641241072084676455533234, −7.73865828747377914951729213009, −7.24443827435431798091802148267, −6.22574815540833714031283437494, −5.27190466845715108765869362211, −4.48258397971251574389629801982, −3.33611157422544257194583862707, −2.53319485263396092629882859053, −0.835548470514578964121677318960, 0.835548470514578964121677318960, 2.53319485263396092629882859053, 3.33611157422544257194583862707, 4.48258397971251574389629801982, 5.27190466845715108765869362211, 6.22574815540833714031283437494, 7.24443827435431798091802148267, 7.73865828747377914951729213009, 8.645837641241072084676455533234, 9.608203759312194453797257789404

Graph of the ZZ-function along the critical line