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Properties

Label 2-3330-1.1-c1-0-54
Degree 22
Conductor 33303330
Sign −1-1
Analytic cond. 26.590126.5901
Root an. cond. 5.156565.15656
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  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 3·11-s + 14-s + 16-s − 3·17-s − 6·19-s − 20-s − 3·22-s − 2·23-s + 25-s + 28-s + 3·29-s + 3·31-s + 32-s − 3·34-s − 35-s − 37-s − 6·38-s − 40-s − 3·41-s − 43-s − 3·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 1.37·19-s − 0.223·20-s − 0.639·22-s − 0.417·23-s + 1/5·25-s + 0.188·28-s + 0.557·29-s + 0.538·31-s + 0.176·32-s − 0.514·34-s − 0.169·35-s − 0.164·37-s − 0.973·38-s − 0.158·40-s − 0.468·41-s − 0.152·43-s − 0.452·44-s + ⋯

Functional equation

Λ(s)=(3330s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3330s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33303330    =    2⋅32⋅5⋅372 \cdot 3^{2} \cdot 5 \cdot 37
Sign: −1-1
Analytic conductor: 26.590126.5901
Root analytic conductor: 5.156565.15656
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3330, ( :1/2), −1)(2,\ 3330,\ (\ :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−T 1 - T
3 1 1
5 1+T 1 + T
37 1+T 1 + T
good7 1−T+pT2 1 - T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
41 1+3T+pT2 1 + 3 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+13T+pT2 1 + 13 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+15T+pT2 1 + 15 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1+pT2 1 + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−18T+pT2 1 - 18 T + p T^{2}
97 1+7T+pT2 1 + 7 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

−8.146451497996909742414741839962, −7.55947946358028496449425566831, −6.56405990058932844390013504676, −6.08056229568137165407273954903, −4.83332190202465636941309325541, −4.63919100492136925221940798677, −3.56154188220255317605428378165, −2.65455170884079945670816110578, −1.73957054101312727200784638700, 0, 1.73957054101312727200784638700, 2.65455170884079945670816110578, 3.56154188220255317605428378165, 4.63919100492136925221940798677, 4.83332190202465636941309325541, 6.08056229568137165407273954903, 6.56405990058932844390013504676, 7.55947946358028496449425566831, 8.146451497996909742414741839962

Graph of the ZZ-function along the critical line