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Properties

Label 2-153-1.1-c3-0-10
Degree 22
Conductor 153153
Sign −1-1
Analytic cond. 9.027299.02729
Root an. cond. 3.004543.00454
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.75·2-s + 14.6·4-s + 7.65·5-s − 31.5·7-s − 31.6·8-s − 36.4·10-s + 7.18·11-s + 84.3·13-s + 150.·14-s + 33.5·16-s − 17·17-s − 37.0·19-s + 112.·20-s − 34.2·22-s − 150.·23-s − 66.3·25-s − 401.·26-s − 462.·28-s + 11.5·29-s − 53.2·31-s + 93.7·32-s + 80.9·34-s − 241.·35-s − 99.2·37-s + 176.·38-s − 242.·40-s − 118.·41-s + ⋯
L(s)  = 1  − 1.68·2-s + 1.83·4-s + 0.684·5-s − 1.70·7-s − 1.40·8-s − 1.15·10-s + 0.197·11-s + 1.79·13-s + 2.87·14-s + 0.524·16-s − 0.242·17-s − 0.447·19-s + 1.25·20-s − 0.331·22-s − 1.36·23-s − 0.531·25-s − 3.02·26-s − 3.12·28-s + 0.0741·29-s − 0.308·31-s + 0.517·32-s + 0.408·34-s − 1.16·35-s − 0.440·37-s + 0.753·38-s − 0.958·40-s − 0.450·41-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: −1-1
Analytic conductor: 9.027299.02729
Root analytic conductor: 3.004543.00454
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 153, ( :3/2), −1)(2,\ 153,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1 1
17 1+17T 1 + 17T
good2 1+4.75T+8T2 1 + 4.75T + 8T^{2}
5 1−7.65T+125T2 1 - 7.65T + 125T^{2}
7 1+31.5T+343T2 1 + 31.5T + 343T^{2}
11 1−7.18T+1.33e3T2 1 - 7.18T + 1.33e3T^{2}
13 1−84.3T+2.19e3T2 1 - 84.3T + 2.19e3T^{2}
19 1+37.0T+6.85e3T2 1 + 37.0T + 6.85e3T^{2}
23 1+150.T+1.21e4T2 1 + 150.T + 1.21e4T^{2}
29 1−11.5T+2.43e4T2 1 - 11.5T + 2.43e4T^{2}
31 1+53.2T+2.97e4T2 1 + 53.2T + 2.97e4T^{2}
37 1+99.2T+5.06e4T2 1 + 99.2T + 5.06e4T^{2}
41 1+118.T+6.89e4T2 1 + 118.T + 6.89e4T^{2}
43 1+456.T+7.95e4T2 1 + 456.T + 7.95e4T^{2}
47 1+571.T+1.03e5T2 1 + 571.T + 1.03e5T^{2}
53 1+462.T+1.48e5T2 1 + 462.T + 1.48e5T^{2}
59 1+48.0T+2.05e5T2 1 + 48.0T + 2.05e5T^{2}
61 1−59.5T+2.26e5T2 1 - 59.5T + 2.26e5T^{2}
67 1+740.T+3.00e5T2 1 + 740.T + 3.00e5T^{2}
71 1−930.T+3.57e5T2 1 - 930.T + 3.57e5T^{2}
73 1+697.T+3.89e5T2 1 + 697.T + 3.89e5T^{2}
79 1−1.03e3T+4.93e5T2 1 - 1.03e3T + 4.93e5T^{2}
83 1−22.2T+5.71e5T2 1 - 22.2T + 5.71e5T^{2}
89 1−369.T+7.04e5T2 1 - 369.T + 7.04e5T^{2}
97 1−1.13e3T+9.12e5T2 1 - 1.13e3T + 9.12e5T^{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

−11.65644661908976076971321866031, −10.50400401685843674619413223268, −9.838401726963965250614617301965, −9.068366168463588903802152142660, −8.142182592671096268013463040669, −6.56917714299354251397802095120, −6.17627826640342648198481797484, −3.47203228415881083087051326614, −1.75646561566549658193007977758, 0, 1.75646561566549658193007977758, 3.47203228415881083087051326614, 6.17627826640342648198481797484, 6.56917714299354251397802095120, 8.142182592671096268013463040669, 9.068366168463588903802152142660, 9.838401726963965250614617301965, 10.50400401685843674619413223268, 11.65644661908976076971321866031

Graph of the ZZ-function along the critical line