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Properties

Label 2-4410-1.1-c1-0-6
Degree 22
Conductor 44104410
Sign 11
Analytic cond. 35.214035.2140
Root an. cond. 5.934145.93414
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s − 2·11-s − 2·13-s + 16-s − 4·17-s + 20-s + 2·22-s − 8·23-s + 25-s + 2·26-s + 2·31-s − 32-s + 4·34-s + 8·37-s − 40-s − 2·41-s − 2·43-s − 2·44-s + 8·46-s + 10·47-s − 50-s − 2·52-s + 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s + 1/4·16-s − 0.970·17-s + 0.223·20-s + 0.426·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.359·31-s − 0.176·32-s + 0.685·34-s + 1.31·37-s − 0.158·40-s − 0.312·41-s − 0.304·43-s − 0.301·44-s + 1.17·46-s + 1.45·47-s − 0.141·50-s − 0.277·52-s + 0.274·53-s + ⋯

Functional equation

Λ(s)=(4410s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4410s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 44104410    =    2⋅32⋅5⋅722 \cdot 3^{2} \cdot 5 \cdot 7^{2}
Sign: 11
Analytic conductor: 35.214035.2140
Root analytic conductor: 5.934145.93414
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4410, ( :1/2), 1)(2,\ 4410,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1133771891.113377189
L(12)L(\frac12) ≈\approx 1.1133771891.113377189
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+T 1 + T
3 1 1
5 1−T 1 - T
7 1 1
good11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1−10T+pT2 1 - 10 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−8.275900421424892036269046670695, −7.81801013988451065415789760990, −6.95159938169996227063425188035, −6.27985274792127521772984106731, −5.55492505029655878101511907953, −4.67596156571065197352692680739, −3.74754759678486478818103175819, −2.48441411623192167811216141892, −2.08868828855827221586844925041, −0.63211088604567526362737749078, 0.63211088604567526362737749078, 2.08868828855827221586844925041, 2.48441411623192167811216141892, 3.74754759678486478818103175819, 4.67596156571065197352692680739, 5.55492505029655878101511907953, 6.27985274792127521772984106731, 6.95159938169996227063425188035, 7.81801013988451065415789760990, 8.275900421424892036269046670695

Graph of the ZZ-function along the critical line