Properties

Label 2-388080-1.1-c1-0-129
Degree $2$
Conductor $388080$
Sign $1$
Analytic cond. $3098.83$
Root an. cond. $55.6671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 11-s + 2·13-s + 6·17-s + 4·19-s + 25-s + 6·29-s + 6·37-s − 10·41-s − 4·43-s − 8·47-s + 6·53-s + 55-s + 8·59-s − 2·61-s − 2·65-s + 8·67-s − 8·71-s + 14·73-s − 4·79-s + 16·83-s − 6·85-s − 10·89-s − 4·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.301·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 0.824·53-s + 0.134·55-s + 1.04·59-s − 0.256·61-s − 0.248·65-s + 0.977·67-s − 0.949·71-s + 1.63·73-s − 0.450·79-s + 1.75·83-s − 0.650·85-s − 1.05·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(388080\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(3098.83\)
Root analytic conductor: \(55.6671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 388080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.059591755\)
\(L(\frac12)\) \(\approx\) \(3.059591755\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.26339136390208, −12.06734696260463, −11.55855936113001, −11.25172597114528, −10.59700504389076, −10.17990566862380, −9.795757581478815, −9.433312338875573, −8.669197605289756, −8.274580003242719, −8.015188447945789, −7.470895567628665, −6.960961351576450, −6.508396546049607, −5.977773080521149, −5.335386757611222, −5.109768118711552, −4.505140659634945, −3.825596077340251, −3.369149709820543, −3.058024733410834, −2.360961291104001, −1.599585223177835, −1.017425251507545, −0.5240316550177075, 0.5240316550177075, 1.017425251507545, 1.599585223177835, 2.360961291104001, 3.058024733410834, 3.369149709820543, 3.825596077340251, 4.505140659634945, 5.109768118711552, 5.335386757611222, 5.977773080521149, 6.508396546049607, 6.960961351576450, 7.470895567628665, 8.015188447945789, 8.274580003242719, 8.669197605289756, 9.433312338875573, 9.795757581478815, 10.17990566862380, 10.59700504389076, 11.25172597114528, 11.55855936113001, 12.06734696260463, 12.26339136390208

Graph of the $Z$-function along the critical line