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Properties

Label 2-3042-1.1-c1-0-48
Degree 22
Conductor 30423042
Sign −1-1
Analytic cond. 24.290424.2904
Root an. cond. 4.928534.92853
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 0.445·5-s + 0.911·7-s − 8-s − 0.445·10-s + 1.64·11-s − 0.911·14-s + 16-s − 1.50·17-s − 1.60·19-s + 0.445·20-s − 1.64·22-s − 7.38·23-s − 4.80·25-s + 0.911·28-s − 5.24·29-s + 7.34·31-s − 32-s + 1.50·34-s + 0.405·35-s − 2.98·37-s + 1.60·38-s − 0.445·40-s − 5.82·41-s + 2.98·43-s + 1.64·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.199·5-s + 0.344·7-s − 0.353·8-s − 0.140·10-s + 0.495·11-s − 0.243·14-s + 0.250·16-s − 0.365·17-s − 0.367·19-s + 0.0995·20-s − 0.350·22-s − 1.53·23-s − 0.960·25-s + 0.172·28-s − 0.974·29-s + 1.31·31-s − 0.176·32-s + 0.258·34-s + 0.0685·35-s − 0.491·37-s + 0.260·38-s − 0.0703·40-s − 0.909·41-s + 0.455·43-s + 0.247·44-s + ⋯

Functional equation

Λ(s)=(3042s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3042s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30423042    =    2⋅32⋅1322 \cdot 3^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 24.290424.2904
Root analytic conductor: 4.928534.92853
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3042, ( :1/2), −1)(2,\ 3042,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T 1 + T
3 1 1
13 1 1
good5 1−0.445T+5T2 1 - 0.445T + 5T^{2}
7 1−0.911T+7T2 1 - 0.911T + 7T^{2}
11 1−1.64T+11T2 1 - 1.64T + 11T^{2}
17 1+1.50T+17T2 1 + 1.50T + 17T^{2}
19 1+1.60T+19T2 1 + 1.60T + 19T^{2}
23 1+7.38T+23T2 1 + 7.38T + 23T^{2}
29 1+5.24T+29T2 1 + 5.24T + 29T^{2}
31 1−7.34T+31T2 1 - 7.34T + 31T^{2}
37 1+2.98T+37T2 1 + 2.98T + 37T^{2}
41 1+5.82T+41T2 1 + 5.82T + 41T^{2}
43 1−2.98T+43T2 1 - 2.98T + 43T^{2}
47 1+7.87T+47T2 1 + 7.87T + 47T^{2}
53 1+10.7T+53T2 1 + 10.7T + 53T^{2}
59 1−10.8T+59T2 1 - 10.8T + 59T^{2}
61 1−6.98T+61T2 1 - 6.98T + 61T^{2}
67 1+6.81T+67T2 1 + 6.81T + 67T^{2}
71 1+6.05T+71T2 1 + 6.05T + 71T^{2}
73 1−8.54T+73T2 1 - 8.54T + 73T^{2}
79 1−6.73T+79T2 1 - 6.73T + 79T^{2}
83 1+3.67T+83T2 1 + 3.67T + 83T^{2}
89 1−16.5T+89T2 1 - 16.5T + 89T^{2}
97 1+5.77T+97T2 1 + 5.77T + 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

−8.187350121472182160683098335503, −7.926017243502113929252447575812, −6.78860496745189447055150633712, −6.27875341334983626250156431973, −5.39885398308011029351786626128, −4.36405644664551466604964885112, −3.51571079692847603514504244347, −2.25911194757770128597110828222, −1.53922091242418604742048138197, 0, 1.53922091242418604742048138197, 2.25911194757770128597110828222, 3.51571079692847603514504244347, 4.36405644664551466604964885112, 5.39885398308011029351786626128, 6.27875341334983626250156431973, 6.78860496745189447055150633712, 7.926017243502113929252447575812, 8.187350121472182160683098335503

Graph of the ZZ-function along the critical line