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Properties

Label 2-450-1.1-c3-0-15
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 + 7-s − 8·8-s − 42·11-s + 67·13-s − 2·14-s + 16·16-s + 54·17-s − 115·19-s + 84·22-s − 162·23-s − 134·26-s + 4·28-s + 210·29-s − 193·31-s − 32·32-s − 108·34-s + 286·37-s + 230·38-s − 12·41-s − 263·43-s − 168·44-s + 324·46-s + 414·47-s − 342·49-s + 268·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.0539·7-s − 0.353·8-s − 1.15·11-s + 1.42·13-s − 0.0381·14-s + 1/4·16-s + 0.770·17-s − 1.38·19-s + 0.814·22-s − 1.46·23-s − 1.01·26-s + 0.0269·28-s + 1.34·29-s − 1.11·31-s − 0.176·32-s − 0.544·34-s + 1.27·37-s + 0.981·38-s − 0.0457·41-s − 0.932·43-s − 0.575·44-s + 1.03·46-s + 1.28·47-s − 0.997·49-s + 0.714·52-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−T+p3T2 1 - T + p^{3} T^{2}
11 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
13 1−67T+p3T2 1 - 67 T + p^{3} T^{2}
17 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
19 1+115T+p3T2 1 + 115 T + p^{3} T^{2}
23 1+162T+p3T2 1 + 162 T + p^{3} T^{2}
29 1−210T+p3T2 1 - 210 T + p^{3} T^{2}
31 1+193T+p3T2 1 + 193 T + p^{3} T^{2}
37 1−286T+p3T2 1 - 286 T + p^{3} T^{2}
41 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
43 1+263T+p3T2 1 + 263 T + p^{3} T^{2}
47 1−414T+p3T2 1 - 414 T + p^{3} T^{2}
53 1+192T+p3T2 1 + 192 T + p^{3} T^{2}
59 1+690T+p3T2 1 + 690 T + p^{3} T^{2}
61 1+733T+p3T2 1 + 733 T + p^{3} T^{2}
67 1+299T+p3T2 1 + 299 T + p^{3} T^{2}
71 1−228T+p3T2 1 - 228 T + p^{3} T^{2}
73 1+938T+p3T2 1 + 938 T + p^{3} T^{2}
79 1+160T+p3T2 1 + 160 T + p^{3} T^{2}
83 1+462T+p3T2 1 + 462 T + p^{3} T^{2}
89 1−240T+p3T2 1 - 240 T + p^{3} T^{2}
97 1−511T+p3T2 1 - 511 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

−10.42742159732335419909950751930, −9.285560660293003633864848810905, −8.261227152270461626443946015735, −7.83585590572544441062544354583, −6.45215667214553317431699458332, −5.72176214258874655195980911260, −4.25678487168865724890900424863, −2.92238207896602842104644539769, −1.57819120348360981361813177911, 0, 1.57819120348360981361813177911, 2.92238207896602842104644539769, 4.25678487168865724890900424863, 5.72176214258874655195980911260, 6.45215667214553317431699458332, 7.83585590572544441062544354583, 8.261227152270461626443946015735, 9.285560660293003633864848810905, 10.42742159732335419909950751930

Graph of the ZZ-function along the critical line