Properties

Label 2-30960-1.1-c1-0-19
Degree $2$
Conductor $30960$
Sign $1$
Analytic cond. $247.216$
Root an. cond. $15.7231$
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 + 4·7-s + 2·13-s + 6·17-s − 2·19-s + 6·23-s + 25-s − 6·29-s + 4·31-s − 4·35-s − 4·37-s − 43-s + 6·47-s + 9·49-s + 6·53-s + 12·59-s − 10·61-s − 2·65-s + 4·67-s − 4·73-s + 4·79-s − 6·83-s − 6·85-s + 6·89-s + 8·91-s + 2·95-s − 10·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s + 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.676·35-s − 0.657·37-s − 0.152·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + 0.488·67-s − 0.468·73-s + 0.450·79-s − 0.658·83-s − 0.650·85-s + 0.635·89-s + 0.838·91-s + 0.205·95-s − 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $1$
Analytic conductor: \(247.216\)
Root analytic conductor: \(15.7231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.192685671\)
\(L(\frac12)\) \(\approx\) \(3.192685671\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
43 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.95563202909817, −14.72945911365083, −14.10136777861829, −13.60979570781375, −12.96930402175846, −12.35785542979178, −11.81661767462185, −11.41804922810108, −10.83228157900626, −10.48407806850573, −9.728702458626000, −8.992439227047922, −8.457545451378919, −8.108015160664720, −7.374919592434173, −7.112445875582425, −6.113242482002973, −5.478071799238423, −5.035813654049411, −4.351421045792665, −3.727554786832462, −3.063300569807305, −2.144868518699847, −1.380849532699156, −0.7536486302293520, 0.7536486302293520, 1.380849532699156, 2.144868518699847, 3.063300569807305, 3.727554786832462, 4.351421045792665, 5.035813654049411, 5.478071799238423, 6.113242482002973, 7.112445875582425, 7.374919592434173, 8.108015160664720, 8.457545451378919, 8.992439227047922, 9.728702458626000, 10.48407806850573, 10.83228157900626, 11.41804922810108, 11.81661767462185, 12.35785542979178, 12.96930402175846, 13.60979570781375, 14.10136777861829, 14.72945911365083, 14.95563202909817

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