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Properties

Label 2-384-1.1-c1-0-7
Degree 22
Conductor 384384
Sign −1-1
Analytic cond. 3.066253.06625
Root an. cond. 1.751071.75107
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  + 3-s − 4·5-s − 2·7-s + 9-s − 4·11-s + 2·13-s − 4·15-s − 2·17-s − 8·19-s − 2·21-s − 4·23-s + 11·25-s + 27-s + 6·31-s − 4·33-s + 8·35-s − 2·37-s + 2·39-s + 6·41-s − 4·45-s − 4·47-s − 3·49-s − 2·51-s + 16·55-s − 8·57-s + 4·59-s + 14·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 1.03·15-s − 0.485·17-s − 1.83·19-s − 0.436·21-s − 0.834·23-s + 11/5·25-s + 0.192·27-s + 1.07·31-s − 0.696·33-s + 1.35·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.596·45-s − 0.583·47-s − 3/7·49-s − 0.280·51-s + 2.15·55-s − 1.05·57-s + 0.520·59-s + 1.79·61-s + ⋯

Functional equation

Λ(s)=(384s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(384s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 384384    =    27⋅32^{7} \cdot 3
Sign: −1-1
Analytic conductor: 3.066253.06625
Root analytic conductor: 1.751071.75107
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 384, ( :1/2), −1)(2,\ 384,\ (\ :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
3 1−T 1 - T
good5 1+4T+pT2 1 + 4 T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+8T+pT2 1 + 8 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−6T+pT2 1 - 6 T + 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+pT2 1 + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+4T+pT2 1 + 4 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+10T+pT2 1 + 10 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−10.86660959821522152324936701040, −10.10337746972486500130129788437, −8.649034097305583887898795668373, −8.230854059399453444196153124887, −7.30623361332829545195635303299, −6.27798985919626463740519529757, −4.56007962912554840821971374527, −3.77214318657172177768570145308, −2.65190583901467307643981603895, 0, 2.65190583901467307643981603895, 3.77214318657172177768570145308, 4.56007962912554840821971374527, 6.27798985919626463740519529757, 7.30623361332829545195635303299, 8.230854059399453444196153124887, 8.649034097305583887898795668373, 10.10337746972486500130129788437, 10.86660959821522152324936701040

Graph of the ZZ-function along the critical line