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Properties

Label 2-42e2-1.1-c3-0-45
Degree 22
Conductor 17641764
Sign −1-1
Analytic cond. 104.079104.079
Root an. cond. 10.201910.2019
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·5-s − 4·11-s − 54·13-s − 14·17-s − 92·19-s + 152·23-s + 71·25-s + 106·29-s + 144·31-s + 158·37-s − 390·41-s − 508·43-s − 528·47-s − 606·53-s − 56·55-s − 364·59-s − 678·61-s − 756·65-s + 844·67-s + 8·71-s + 422·73-s + 384·79-s − 548·83-s − 196·85-s + 1.19e3·89-s − 1.28e3·95-s + 1.50e3·97-s + ⋯
L(s)  = 1  + 1.25·5-s − 0.109·11-s − 1.15·13-s − 0.199·17-s − 1.11·19-s + 1.37·23-s + 0.567·25-s + 0.678·29-s + 0.834·31-s + 0.702·37-s − 1.48·41-s − 1.80·43-s − 1.63·47-s − 1.57·53-s − 0.137·55-s − 0.803·59-s − 1.42·61-s − 1.44·65-s + 1.53·67-s + 0.0133·71-s + 0.676·73-s + 0.546·79-s − 0.724·83-s − 0.250·85-s + 1.42·89-s − 1.39·95-s + 1.57·97-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 104.079104.079
Root analytic conductor: 10.201910.2019
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1764, ( :3/2), −1)(2,\ 1764,\ (\ :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)
bad2 1 1
3 1 1
7 1 1
good5 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
11 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
13 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
17 1+14T+p3T2 1 + 14 T + p^{3} T^{2}
19 1+92T+p3T2 1 + 92 T + p^{3} T^{2}
23 1−152T+p3T2 1 - 152 T + p^{3} T^{2}
29 1−106T+p3T2 1 - 106 T + p^{3} T^{2}
31 1−144T+p3T2 1 - 144 T + p^{3} T^{2}
37 1−158T+p3T2 1 - 158 T + p^{3} T^{2}
41 1+390T+p3T2 1 + 390 T + p^{3} T^{2}
43 1+508T+p3T2 1 + 508 T + p^{3} T^{2}
47 1+528T+p3T2 1 + 528 T + p^{3} T^{2}
53 1+606T+p3T2 1 + 606 T + p^{3} T^{2}
59 1+364T+p3T2 1 + 364 T + p^{3} T^{2}
61 1+678T+p3T2 1 + 678 T + p^{3} T^{2}
67 1−844T+p3T2 1 - 844 T + p^{3} T^{2}
71 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
73 1−422T+p3T2 1 - 422 T + p^{3} T^{2}
79 1−384T+p3T2 1 - 384 T + p^{3} T^{2}
83 1+548T+p3T2 1 + 548 T + p^{3} T^{2}
89 1−1194T+p3T2 1 - 1194 T + p^{3} T^{2}
97 1−1502T+p3T2 1 - 1502 T + p^{3} 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

−8.614467037088178232126517842165, −7.81935535784415108227138387633, −6.59279040186920272429682738196, −6.41038534571295420806227146525, −5.04589330251369409521972065992, −4.79236387925037506460691787587, −3.22268254652689818743861051328, −2.35862291006816767023424067526, −1.48069920034264539280955165435, 0, 1.48069920034264539280955165435, 2.35862291006816767023424067526, 3.22268254652689818743861051328, 4.79236387925037506460691787587, 5.04589330251369409521972065992, 6.41038534571295420806227146525, 6.59279040186920272429682738196, 7.81935535784415108227138387633, 8.614467037088178232126517842165

Graph of the ZZ-function along the critical line