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Properties

Label 2-440-1.1-c1-0-9
Degree 22
Conductor 440440
Sign −1-1
Analytic cond. 3.513413.51341
Root an. cond. 1.874411.87441
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 3·9-s + 11-s − 4·13-s − 4·17-s + 25-s − 6·29-s + 2·35-s − 2·37-s + 6·41-s + 2·43-s + 3·45-s − 3·49-s − 10·53-s − 55-s + 12·59-s − 6·61-s + 6·63-s + 4·65-s − 12·67-s + 16·71-s + 4·73-s − 2·77-s − 4·79-s + 9·81-s + 2·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 9-s + 0.301·11-s − 1.10·13-s − 0.970·17-s + 1/5·25-s − 1.11·29-s + 0.338·35-s − 0.328·37-s + 0.937·41-s + 0.304·43-s + 0.447·45-s − 3/7·49-s − 1.37·53-s − 0.134·55-s + 1.56·59-s − 0.768·61-s + 0.755·63-s + 0.496·65-s − 1.46·67-s + 1.89·71-s + 0.468·73-s − 0.227·77-s − 0.450·79-s + 81-s + 0.219·83-s + ⋯

Functional equation

Λ(s)=(440s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(440s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 440440    =    23⋅5⋅112^{3} \cdot 5 \cdot 11
Sign: −1-1
Analytic conductor: 3.513413.51341
Root analytic conductor: 1.874411.87441
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 440, ( :1/2), −1)(2,\ 440,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
5 1+T 1 + T
11 1−T 1 - T
good3 1+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−16T+pT2 1 - 16 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+2T+pT2 1 + 2 T + p 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

−10.81445160960969925806257181434, −9.620258313556316476955944146421, −8.987128940086315118720924933355, −7.906393321228809348111883565536, −6.94257154461568951821487101757, −5.99500771172096974750229105298, −4.82772136865313429170223708877, −3.58711967186630926697460909520, −2.44118808260323168492313335198, 0, 2.44118808260323168492313335198, 3.58711967186630926697460909520, 4.82772136865313429170223708877, 5.99500771172096974750229105298, 6.94257154461568951821487101757, 7.906393321228809348111883565536, 8.987128940086315118720924933355, 9.620258313556316476955944146421, 10.81445160960969925806257181434

Graph of the ZZ-function along the critical line