Properties

Degree $2$
Conductor $4032$
Sign $1$
Motivic weight $1$
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 − 6·13-s − 2·17-s − 4·19-s + 8·23-s − 25-s − 2·29-s − 2·35-s + 10·37-s + 6·41-s − 4·43-s + 49-s + 6·53-s − 8·55-s − 4·59-s − 6·61-s + 12·65-s + 4·67-s + 8·71-s + 10·73-s + 4·77-s + 4·83-s + 4·85-s + 6·89-s − 6·91-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 1.20·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.371·29-s − 0.338·35-s + 1.64·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s − 1.07·55-s − 0.520·59-s − 0.768·61-s + 1.48·65-s + 0.488·67-s + 0.949·71-s + 1.17·73-s + 0.455·77-s + 0.439·83-s + 0.433·85-s + 0.635·89-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: $1$
Motivic weight: \(1\)
Character: $\chi_{4032} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.418845483\)
\(L(\frac12)\) \(\approx\) \(1.418845483\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
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 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−18.03762756715260, −17.34674386537838, −16.86461218402957, −16.50017169365781, −15.37769240296599, −14.96621205207054, −14.68598176721100, −13.85948642424766, −12.96840492476808, −12.40878376259858, −11.80956405785084, −11.22075297586092, −10.72252892462507, −9.575429059410093, −9.258735862864324, −8.357312414327508, −7.687138909555049, −7.037619494193415, −6.437916833982970, −5.322401419345924, −4.490916403668538, −4.068024435551992, −2.950754869354144, −2.013763634717605, −0.6738920945868243, 0.6738920945868243, 2.013763634717605, 2.950754869354144, 4.068024435551992, 4.490916403668538, 5.322401419345924, 6.437916833982970, 7.037619494193415, 7.687138909555049, 8.357312414327508, 9.258735862864324, 9.575429059410093, 10.72252892462507, 11.22075297586092, 11.80956405785084, 12.40878376259858, 12.96840492476808, 13.85948642424766, 14.68598176721100, 14.96621205207054, 15.37769240296599, 16.50017169365781, 16.86461218402957, 17.34674386537838, 18.03762756715260

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