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Properties

Label 2-39e2-1.1-c3-0-19
Degree 22
Conductor 15211521
Sign 11
Analytic cond. 89.741989.7419
Root an. cond. 9.473229.47322
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s − 20·7-s + 64·16-s − 56·19-s − 125·25-s + 160·28-s − 308·31-s − 110·37-s − 520·43-s + 57·49-s + 182·61-s − 512·64-s + 880·67-s − 1.19e3·73-s + 448·76-s + 884·79-s + 1.33e3·97-s + 1.00e3·100-s + 1.82e3·103-s + 646·109-s − 1.28e3·112-s + ⋯
L(s)  = 1  − 4-s − 1.07·7-s + 16-s − 0.676·19-s − 25-s + 1.07·28-s − 1.78·31-s − 0.488·37-s − 1.84·43-s + 0.166·49-s + 0.382·61-s − 64-s + 1.60·67-s − 1.90·73-s + 0.676·76-s + 1.25·79-s + 1.39·97-s + 100-s + 1.74·103-s + 0.567·109-s − 1.07·112-s + ⋯

Functional equation

Λ(s)=(1521s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1521s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15211521    =    32⋅1323^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 89.741989.7419
Root analytic conductor: 9.473229.47322
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1521, ( :3/2), 1)(2,\ 1521,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.58368897780.5836889778
L(12)L(\frac12) ≈\approx 0.58368897780.5836889778
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)
bad3 1 1
13 1 1
good2 1+p3T2 1 + p^{3} T^{2}
5 1+p3T2 1 + p^{3} T^{2}
7 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
11 1+p3T2 1 + p^{3} T^{2}
17 1+p3T2 1 + p^{3} T^{2}
19 1+56T+p3T2 1 + 56 T + p^{3} T^{2}
23 1+p3T2 1 + p^{3} T^{2}
29 1+p3T2 1 + p^{3} T^{2}
31 1+308T+p3T2 1 + 308 T + p^{3} T^{2}
37 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
41 1+p3T2 1 + p^{3} T^{2}
43 1+520T+p3T2 1 + 520 T + p^{3} T^{2}
47 1+p3T2 1 + p^{3} T^{2}
53 1+p3T2 1 + p^{3} T^{2}
59 1+p3T2 1 + p^{3} T^{2}
61 1−182T+p3T2 1 - 182 T + p^{3} T^{2}
67 1−880T+p3T2 1 - 880 T + p^{3} T^{2}
71 1+p3T2 1 + p^{3} T^{2}
73 1+1190T+p3T2 1 + 1190 T + p^{3} T^{2}
79 1−884T+p3T2 1 - 884 T + p^{3} T^{2}
83 1+p3T2 1 + p^{3} T^{2}
89 1+p3T2 1 + p^{3} T^{2}
97 1−1330T+p3T2 1 - 1330 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

−9.128259920149454599105881726584, −8.474264327392467569383426992496, −7.55326526238465471383452984841, −6.60824334894450726365154115399, −5.78491532683739033575566513445, −4.94986190520765637234160228414, −3.88153215661751968936986651283, −3.31385040095635340929951606062, −1.88323948183718506439718852949, −0.36382943752534658890646139082, 0.36382943752534658890646139082, 1.88323948183718506439718852949, 3.31385040095635340929951606062, 3.88153215661751968936986651283, 4.94986190520765637234160228414, 5.78491532683739033575566513445, 6.60824334894450726365154115399, 7.55326526238465471383452984841, 8.474264327392467569383426992496, 9.128259920149454599105881726584

Graph of the ZZ-function along the critical line