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Properties

Label 2-3042-1.1-c1-0-50
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 − 4.04·5-s + 0.692·7-s + 8-s − 4.04·10-s + 4.85·11-s + 0.692·14-s + 16-s − 7.38·17-s − 1.78·19-s − 4.04·20-s + 4.85·22-s − 5.10·23-s + 11.3·25-s + 0.692·28-s + 3.34·29-s + 0.972·31-s + 32-s − 7.38·34-s − 2.80·35-s + 1.28·37-s − 1.78·38-s − 4.04·40-s − 1.50·41-s − 8.31·43-s + 4.85·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.81·5-s + 0.261·7-s + 0.353·8-s − 1.28·10-s + 1.46·11-s + 0.184·14-s + 0.250·16-s − 1.79·17-s − 0.408·19-s − 0.905·20-s + 1.03·22-s − 1.06·23-s + 2.27·25-s + 0.130·28-s + 0.621·29-s + 0.174·31-s + 0.176·32-s − 1.26·34-s − 0.473·35-s + 0.211·37-s − 0.288·38-s − 0.640·40-s − 0.235·41-s − 1.26·43-s + 0.731·44-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+4.04T+5T2 1 + 4.04T + 5T^{2}
7 1−0.692T+7T2 1 - 0.692T + 7T^{2}
11 1−4.85T+11T2 1 - 4.85T + 11T^{2}
17 1+7.38T+17T2 1 + 7.38T + 17T^{2}
19 1+1.78T+19T2 1 + 1.78T + 19T^{2}
23 1+5.10T+23T2 1 + 5.10T + 23T^{2}
29 1−3.34T+29T2 1 - 3.34T + 29T^{2}
31 1−0.972T+31T2 1 - 0.972T + 31T^{2}
37 1−1.28T+37T2 1 - 1.28T + 37T^{2}
41 1+1.50T+41T2 1 + 1.50T + 41T^{2}
43 1+8.31T+43T2 1 + 8.31T + 43T^{2}
47 1−7.20T+47T2 1 - 7.20T + 47T^{2}
53 1+13.4T+53T2 1 + 13.4T + 53T^{2}
59 1+1.30T+59T2 1 + 1.30T + 59T^{2}
61 1+0.396T+61T2 1 + 0.396T + 61T^{2}
67 1−6.05T+67T2 1 - 6.05T + 67T^{2}
71 1−1.32T+71T2 1 - 1.32T + 71T^{2}
73 1+7.65T+73T2 1 + 7.65T + 73T^{2}
79 1+8.33T+79T2 1 + 8.33T + 79T^{2}
83 1+15.3T+83T2 1 + 15.3T + 83T^{2}
89 1+3.10T+89T2 1 + 3.10T + 89T^{2}
97 1+8.54T+97T2 1 + 8.54T + 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

−8.361857692340964722875491255891, −7.50719277773770057838819113870, −6.71177688853144678335199103810, −6.31265329841930805232147794356, −4.88771208837074146954305591760, −4.23204533733098603904573318429, −3.90681291207331497293349134746, −2.86887352156986316756235186105, −1.56759099700760226058139542948, 0, 1.56759099700760226058139542948, 2.86887352156986316756235186105, 3.90681291207331497293349134746, 4.23204533733098603904573318429, 4.88771208837074146954305591760, 6.31265329841930805232147794356, 6.71177688853144678335199103810, 7.50719277773770057838819113870, 8.361857692340964722875491255891

Graph of the ZZ-function along the critical line