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Properties

Label 2-450-1.1-c3-0-23
Degree 22
Conductor 450450
Sign −1-1
Analytic cond. 26.550826.5508
Root an. cond. 5.152755.15275
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  + 2·2-s + 4·4-s + 2·7-s + 8·8-s − 70·11-s − 54·13-s + 4·14-s + 16·16-s − 22·17-s + 24·19-s − 140·22-s − 100·23-s − 108·26-s + 8·28-s − 216·29-s + 208·31-s + 32·32-s − 44·34-s + 254·37-s + 48·38-s + 206·41-s − 292·43-s − 280·44-s − 200·46-s − 320·47-s − 339·49-s − 216·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.107·7-s + 0.353·8-s − 1.91·11-s − 1.15·13-s + 0.0763·14-s + 1/4·16-s − 0.313·17-s + 0.289·19-s − 1.35·22-s − 0.906·23-s − 0.814·26-s + 0.0539·28-s − 1.38·29-s + 1.20·31-s + 0.176·32-s − 0.221·34-s + 1.12·37-s + 0.204·38-s + 0.784·41-s − 1.03·43-s − 0.959·44-s − 0.641·46-s − 0.993·47-s − 0.988·49-s − 0.576·52-s + ⋯

Functional equation

Λ(s)=(450s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(450s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 450450    =    2⋅32⋅522 \cdot 3^{2} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 26.550826.5508
Root analytic conductor: 5.152755.15275
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 450, ( :3/2), −1)(2,\ 450,\ (\ :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−pT 1 - p T
3 1 1
5 1 1
good7 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
11 1+70T+p3T2 1 + 70 T + p^{3} T^{2}
13 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
17 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
19 1−24T+p3T2 1 - 24 T + p^{3} T^{2}
23 1+100T+p3T2 1 + 100 T + p^{3} T^{2}
29 1+216T+p3T2 1 + 216 T + p^{3} T^{2}
31 1−208T+p3T2 1 - 208 T + p^{3} T^{2}
37 1−254T+p3T2 1 - 254 T + p^{3} T^{2}
41 1−206T+p3T2 1 - 206 T + p^{3} T^{2}
43 1+292T+p3T2 1 + 292 T + p^{3} T^{2}
47 1+320T+p3T2 1 + 320 T + p^{3} T^{2}
53 1+402T+p3T2 1 + 402 T + p^{3} T^{2}
59 1−370T+p3T2 1 - 370 T + p^{3} T^{2}
61 1+550T+p3T2 1 + 550 T + p^{3} T^{2}
67 1+728T+p3T2 1 + 728 T + p^{3} T^{2}
71 1−540T+p3T2 1 - 540 T + p^{3} T^{2}
73 1+604T+p3T2 1 + 604 T + p^{3} T^{2}
79 1−792T+p3T2 1 - 792 T + p^{3} T^{2}
83 1−404T+p3T2 1 - 404 T + p^{3} T^{2}
89 1−938T+p3T2 1 - 938 T + p^{3} T^{2}
97 1+56T+p3T2 1 + 56 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

−10.30738529281219340529384958067, −9.565781803646040750201508407263, −8.002620034804854150153640753093, −7.58895092690028014822206377155, −6.29073839751339634421066641982, −5.25469291660954935325258950477, −4.55914480399969525270586930644, −3.06334673250803879995167419470, −2.12825028786074897279987242645, 0, 2.12825028786074897279987242645, 3.06334673250803879995167419470, 4.55914480399969525270586930644, 5.25469291660954935325258950477, 6.29073839751339634421066641982, 7.58895092690028014822206377155, 8.002620034804854150153640753093, 9.565781803646040750201508407263, 10.30738529281219340529384958067

Graph of the ZZ-function along the critical line