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Properties

Label 2-2205-1.1-c3-0-113
Degree 22
Conductor 22052205
Sign −1-1
Analytic cond. 130.099130.099
Root an. cond. 11.406111.4061
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  − 5·2-s + 17·4-s + 5·5-s − 45·8-s − 25·10-s − 12·11-s − 30·13-s + 89·16-s − 134·17-s + 92·19-s + 85·20-s + 60·22-s − 112·23-s + 25·25-s + 150·26-s + 58·29-s + 224·31-s − 85·32-s + 670·34-s − 146·37-s − 460·38-s − 225·40-s + 18·41-s + 340·43-s − 204·44-s + 560·46-s + 208·47-s + ⋯
L(s)  = 1  − 1.76·2-s + 17/8·4-s + 0.447·5-s − 1.98·8-s − 0.790·10-s − 0.328·11-s − 0.640·13-s + 1.39·16-s − 1.91·17-s + 1.11·19-s + 0.950·20-s + 0.581·22-s − 1.01·23-s + 1/5·25-s + 1.13·26-s + 0.371·29-s + 1.29·31-s − 0.469·32-s + 3.37·34-s − 0.648·37-s − 1.96·38-s − 0.889·40-s + 0.0685·41-s + 1.20·43-s − 0.698·44-s + 1.79·46-s + 0.645·47-s + ⋯

Functional equation

Λ(s)=(2205s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2205s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22052205    =    32⋅5⋅723^{2} \cdot 5 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 130.099130.099
Root analytic conductor: 11.406111.4061
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2205, ( :3/2), −1)(2,\ 2205,\ (\ :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
5 1−pT 1 - p T
7 1 1
good2 1+5T+p3T2 1 + 5 T + p^{3} T^{2}
11 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
13 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
17 1+134T+p3T2 1 + 134 T + p^{3} T^{2}
19 1−92T+p3T2 1 - 92 T + p^{3} T^{2}
23 1+112T+p3T2 1 + 112 T + p^{3} T^{2}
29 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
31 1−224T+p3T2 1 - 224 T + p^{3} T^{2}
37 1+146T+p3T2 1 + 146 T + p^{3} T^{2}
41 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
43 1−340T+p3T2 1 - 340 T + p^{3} T^{2}
47 1−208T+p3T2 1 - 208 T + p^{3} T^{2}
53 1−754T+p3T2 1 - 754 T + p^{3} T^{2}
59 1−380T+p3T2 1 - 380 T + p^{3} T^{2}
61 1+718T+p3T2 1 + 718 T + p^{3} T^{2}
67 1−412T+p3T2 1 - 412 T + p^{3} T^{2}
71 1−960T+p3T2 1 - 960 T + p^{3} T^{2}
73 1+1066T+p3T2 1 + 1066 T + p^{3} T^{2}
79 1−896T+p3T2 1 - 896 T + p^{3} T^{2}
83 1−436T+p3T2 1 - 436 T + p^{3} T^{2}
89 1+1038T+p3T2 1 + 1038 T + p^{3} T^{2}
97 1−702T+p3T2 1 - 702 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

−8.454518633409177646001690614370, −7.71382454533533323090619164101, −6.97601987220631453432172758318, −6.35940570680834285867658930978, −5.34970623623319497523255948157, −4.22629437241321382356219299771, −2.66196910331395592679785324185, −2.18193678356828036527579923922, −1.00192242183602422735834273132, 0, 1.00192242183602422735834273132, 2.18193678356828036527579923922, 2.66196910331395592679785324185, 4.22629437241321382356219299771, 5.34970623623319497523255948157, 6.35940570680834285867658930978, 6.97601987220631453432172758318, 7.71382454533533323090619164101, 8.454518633409177646001690614370

Graph of the ZZ-function along the critical line