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Properties

Label 4-3400e2-1.1-c0e2-0-8
Degree $4$
Conductor $11560000$
Sign $1$
Analytic cond. $2.87920$
Root an. cond. $1.30262$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 2·9-s + 2·11-s − 6·12-s + 5·16-s + 2·17-s + 4·18-s + 4·22-s − 8·24-s − 2·27-s + 6·32-s − 4·33-s + 4·34-s + 6·36-s − 2·41-s + 6·44-s − 10·48-s − 4·51-s − 4·54-s + 7·64-s − 8·66-s + 6·68-s + 8·72-s + ⋯
L(s)  = 1  + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 2·9-s + 2·11-s − 6·12-s + 5·16-s + 2·17-s + 4·18-s + 4·22-s − 8·24-s − 2·27-s + 6·32-s − 4·33-s + 4·34-s + 6·36-s − 2·41-s + 6·44-s − 10·48-s − 4·51-s − 4·54-s + 7·64-s − 8·66-s + 6·68-s + 8·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11560000\)    =    \(2^{6} \cdot 5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2.87920\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11560000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.784083959\)
\(L(\frac12)\) \(\approx\) \(3.784083959\)
\(L(1)\) 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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−9.003011315011733286897202263854, −8.431224682150013529108607587560, −7.999414654764499291161302299081, −7.53978018297950376832477941522, −7.09976776313948135979500678531, −6.95814448442209925446577500405, −6.34804753083369111674632538790, −6.34565354833593687130139091752, −5.78855039686796265003335310973, −5.65778406117974507217775836602, −5.14780402585633515496536468858, −5.01108322667315234370289882424, −4.35438413167123921694512578717, −4.11341654229099925768809742684, −3.56448364837946694602672571706, −3.39768213304967656533991773737, −2.76580015199538129272136906220, −1.90726839944321565253541722163, −1.39218812189018332306246594231, −1.12651626377619686095662212557, 1.12651626377619686095662212557, 1.39218812189018332306246594231, 1.90726839944321565253541722163, 2.76580015199538129272136906220, 3.39768213304967656533991773737, 3.56448364837946694602672571706, 4.11341654229099925768809742684, 4.35438413167123921694512578717, 5.01108322667315234370289882424, 5.14780402585633515496536468858, 5.65778406117974507217775836602, 5.78855039686796265003335310973, 6.34565354833593687130139091752, 6.34804753083369111674632538790, 6.95814448442209925446577500405, 7.09976776313948135979500678531, 7.53978018297950376832477941522, 7.999414654764499291161302299081, 8.431224682150013529108607587560, 9.003011315011733286897202263854

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