Properties

Label 2-4032-1.1-c1-0-14
Degree $2$
Conductor $4032$
Sign $1$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
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  + 2·5-s − 7-s − 4·11-s − 2·13-s + 6·17-s − 8·19-s − 25-s + 6·29-s + 8·31-s − 2·35-s + 2·37-s − 2·41-s + 4·43-s + 8·47-s + 49-s + 6·53-s − 8·55-s + 6·61-s − 4·65-s + 4·67-s + 8·71-s + 10·73-s + 4·77-s + 16·79-s + 8·83-s + 12·85-s + 6·89-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 1.20·11-s − 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s + 0.768·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s + 1.80·79-s + 0.878·83-s + 1.30·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4032} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.874848804\)
\(L(\frac12)\) \(\approx\) \(1.874848804\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.272203299109375537884659130698, −7.902596514850714806179317231692, −6.84383926850302326577754552348, −6.20669276463558444353009673367, −5.50140758689236696304146355325, −4.83297349612888492276603651901, −3.83348667018088193321396729321, −2.66451378084199854982327930143, −2.24055198785635289229441777506, −0.76540944404158607646425160976, 0.76540944404158607646425160976, 2.24055198785635289229441777506, 2.66451378084199854982327930143, 3.83348667018088193321396729321, 4.83297349612888492276603651901, 5.50140758689236696304146355325, 6.20669276463558444353009673367, 6.84383926850302326577754552348, 7.902596514850714806179317231692, 8.272203299109375537884659130698

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