Properties

Label 2-4032-1.1-c1-0-33
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  + 4·5-s − 7-s + 2·11-s + 6·13-s + 4·17-s + 4·19-s − 2·23-s + 11·25-s − 2·29-s − 4·35-s − 2·37-s + 4·43-s − 12·47-s + 49-s − 6·53-s + 8·55-s − 8·59-s − 6·61-s + 24·65-s + 8·67-s − 14·71-s − 2·73-s − 2·77-s + 12·79-s − 4·83-s + 16·85-s − 6·91-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.377·7-s + 0.603·11-s + 1.66·13-s + 0.970·17-s + 0.917·19-s − 0.417·23-s + 11/5·25-s − 0.371·29-s − 0.676·35-s − 0.328·37-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 1.04·59-s − 0.768·61-s + 2.97·65-s + 0.977·67-s − 1.66·71-s − 0.234·73-s − 0.227·77-s + 1.35·79-s − 0.439·83-s + 1.73·85-s − 0.628·91-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

\(L(1)\) \(\approx\) \(3.210132407\)
\(L(\frac12)\) \(\approx\) \(3.210132407\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−8.629751576240571797976949670652, −7.72604486612696592247209656788, −6.71189699012874428808268514597, −6.09209328369221800215319890295, −5.74755430914602536337090529880, −4.86265553633570999718490113346, −3.63611416585453122580621062957, −3.00487505240709347914613380548, −1.75098525765288305362479637690, −1.17710716193186405280124036592, 1.17710716193186405280124036592, 1.75098525765288305362479637690, 3.00487505240709347914613380548, 3.63611416585453122580621062957, 4.86265553633570999718490113346, 5.74755430914602536337090529880, 6.09209328369221800215319890295, 6.71189699012874428808268514597, 7.72604486612696592247209656788, 8.629751576240571797976949670652

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