Properties

Label 2-4032-1.1-c1-0-16
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  − 7-s + 4·11-s − 2·13-s + 4·17-s − 4·23-s − 5·25-s + 4·29-s + 8·31-s + 2·37-s − 4·41-s + 8·43-s − 8·47-s + 49-s − 4·53-s + 8·59-s + 2·61-s + 8·67-s − 12·71-s + 6·73-s − 4·77-s + 8·79-s + 16·83-s − 12·89-s + 2·91-s − 2·97-s + 8·103-s + 4·107-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.20·11-s − 0.554·13-s + 0.970·17-s − 0.834·23-s − 25-s + 0.742·29-s + 1.43·31-s + 0.328·37-s − 0.624·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.549·53-s + 1.04·59-s + 0.256·61-s + 0.977·67-s − 1.42·71-s + 0.702·73-s − 0.455·77-s + 0.900·79-s + 1.75·83-s − 1.27·89-s + 0.209·91-s − 0.203·97-s + 0.788·103-s + 0.386·107-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.888429370\)
\(L(\frac12)\) \(\approx\) \(1.888429370\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−8.312800947259352999029654855028, −7.82657075768956775624561038408, −6.85455547345968172173263225562, −6.30976897034687746205351939386, −5.56893022235200886998375054341, −4.57255409210211494302382224210, −3.85337102430247200397131844284, −3.01871027790884328597915787157, −1.95048060242068047806603121302, −0.799812534367287661440970892005, 0.799812534367287661440970892005, 1.95048060242068047806603121302, 3.01871027790884328597915787157, 3.85337102430247200397131844284, 4.57255409210211494302382224210, 5.56893022235200886998375054341, 6.30976897034687746205351939386, 6.85455547345968172173263225562, 7.82657075768956775624561038408, 8.312800947259352999029654855028

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