Properties

Degree $4$
Conductor $3969$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 14·7-s + 164·13-s − 55·16-s − 40·19-s − 174·25-s − 42·28-s + 312·31-s + 372·37-s + 328·43-s + 147·49-s + 492·52-s + 1.58e3·61-s − 357·64-s − 88·67-s + 252·73-s − 120·76-s − 1.42e3·79-s − 2.29e3·91-s + 1.59e3·97-s − 522·100-s − 1.83e3·103-s − 684·109-s + 770·112-s − 762·121-s + 936·124-s + 127-s + ⋯
L(s)  = 1  + 3/8·4-s − 0.755·7-s + 3.49·13-s − 0.859·16-s − 0.482·19-s − 1.39·25-s − 0.283·28-s + 1.80·31-s + 1.65·37-s + 1.16·43-s + 3/7·49-s + 1.31·52-s + 3.31·61-s − 0.697·64-s − 0.160·67-s + 0.404·73-s − 0.181·76-s − 2.02·79-s − 2.64·91-s + 1.67·97-s − 0.521·100-s − 1.75·103-s − 0.601·109-s + 0.649·112-s − 0.572·121-s + 0.677·124-s + 0.000698·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3969\)    =    \(3^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3969,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.95863\)
\(L(\frac12)\) \(\approx\) \(1.95863\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
7$C_1$ \( ( 1 + p T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 174 T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 762 T^{2} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 82 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 3670 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 20 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 7234 T^{2} + p^{6} T^{4} \)
29$C_2^2$ \( 1 - 10806 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 - 156 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 186 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 110406 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 - 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 13970 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 273130 T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 386134 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 790 T + p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 44 T + p^{3} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 518146 T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 - 126 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 712 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 1001450 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 - 709626 T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 798 T + p^{3} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.81193538926328242463668602025, −13.96785455537949929393881666455, −13.50640275149007135629872319687, −13.25832281137741270350957042844, −12.69989022722962218748552327117, −11.70011613600586416333818902959, −11.33541398420770267525710753026, −10.96077700243480736518978842590, −10.18051851977067388998539362168, −9.597573342119298833953437372318, −8.679428840177077893829441814286, −8.498283235177754701987777796728, −7.62259836487164578713123781198, −6.48476799588701545015873416061, −6.32883276587383765563298045337, −5.70392303032643907704309463931, −4.18295387428861661860873926800, −3.74253868280017716747034039204, −2.54156397125768629133131145321, −1.07436093738842648879835237771, 1.07436093738842648879835237771, 2.54156397125768629133131145321, 3.74253868280017716747034039204, 4.18295387428861661860873926800, 5.70392303032643907704309463931, 6.32883276587383765563298045337, 6.48476799588701545015873416061, 7.62259836487164578713123781198, 8.498283235177754701987777796728, 8.679428840177077893829441814286, 9.597573342119298833953437372318, 10.18051851977067388998539362168, 10.96077700243480736518978842590, 11.33541398420770267525710753026, 11.70011613600586416333818902959, 12.69989022722962218748552327117, 13.25832281137741270350957042844, 13.50640275149007135629872319687, 13.96785455537949929393881666455, 14.81193538926328242463668602025

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