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Properties

Label 2-1512-1.1-c1-0-5
Degree 22
Conductor 15121512
Sign 11
Analytic cond. 12.073312.0733
Root an. cond. 3.474673.47467
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.37·5-s + 7-s − 1.37·13-s + 17-s + 6.74·19-s − 5.37·23-s + 0.627·25-s + 0.627·29-s + 2.62·31-s − 2.37·35-s + 4.37·37-s + 5.11·41-s + 7.74·43-s + 0.372·47-s + 49-s + 12.1·53-s − 0.255·59-s + 1.25·61-s + 3.25·65-s + 4.62·67-s − 7.37·71-s + 10·73-s − 9.11·79-s + 15.8·83-s − 2.37·85-s + 2.62·89-s − 1.37·91-s + ⋯
L(s)  = 1  − 1.06·5-s + 0.377·7-s − 0.380·13-s + 0.242·17-s + 1.54·19-s − 1.12·23-s + 0.125·25-s + 0.116·29-s + 0.471·31-s − 0.400·35-s + 0.718·37-s + 0.799·41-s + 1.18·43-s + 0.0543·47-s + 0.142·49-s + 1.66·53-s − 0.0332·59-s + 0.160·61-s + 0.403·65-s + 0.565·67-s − 0.874·71-s + 1.17·73-s − 1.02·79-s + 1.74·83-s − 0.257·85-s + 0.278·89-s − 0.143·91-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: no
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 1.3774057781.377405778
L(12)L(\frac12) ≈\approx 1.3774057781.377405778
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+2.37T+5T2 1 + 2.37T + 5T^{2}
11 1+11T2 1 + 11T^{2}
13 1+1.37T+13T2 1 + 1.37T + 13T^{2}
17 1−T+17T2 1 - T + 17T^{2}
19 1−6.74T+19T2 1 - 6.74T + 19T^{2}
23 1+5.37T+23T2 1 + 5.37T + 23T^{2}
29 1−0.627T+29T2 1 - 0.627T + 29T^{2}
31 1−2.62T+31T2 1 - 2.62T + 31T^{2}
37 1−4.37T+37T2 1 - 4.37T + 37T^{2}
41 1−5.11T+41T2 1 - 5.11T + 41T^{2}
43 1−7.74T+43T2 1 - 7.74T + 43T^{2}
47 1−0.372T+47T2 1 - 0.372T + 47T^{2}
53 1−12.1T+53T2 1 - 12.1T + 53T^{2}
59 1+0.255T+59T2 1 + 0.255T + 59T^{2}
61 1−1.25T+61T2 1 - 1.25T + 61T^{2}
67 1−4.62T+67T2 1 - 4.62T + 67T^{2}
71 1+7.37T+71T2 1 + 7.37T + 71T^{2}
73 1−10T+73T2 1 - 10T + 73T^{2}
79 1+9.11T+79T2 1 + 9.11T + 79T^{2}
83 1−15.8T+83T2 1 - 15.8T + 83T^{2}
89 1−2.62T+89T2 1 - 2.62T + 89T^{2}
97 1−9.48T+97T2 1 - 9.48T + 97T^{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

−9.494371253326006661345522740413, −8.558753907729200267060310096879, −7.61484721121862250599219428100, −7.50519700112715179141567584728, −6.18241795578000492124680060514, −5.27426583044841362411508059967, −4.32669599270316909629217427887, −3.57178027611384620479456878112, −2.41648034852488218705625360947, −0.842275583713282463176159217319, 0.842275583713282463176159217319, 2.41648034852488218705625360947, 3.57178027611384620479456878112, 4.32669599270316909629217427887, 5.27426583044841362411508059967, 6.18241795578000492124680060514, 7.50519700112715179141567584728, 7.61484721121862250599219428100, 8.558753907729200267060310096879, 9.494371253326006661345522740413

Graph of the ZZ-function along the critical line