Properties

Label 4-1040e2-1.1-c0e2-0-2
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $0.269389$
Root an. cond. $0.720435$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 2·13-s + 2·17-s + 3·25-s + 2·41-s − 2·49-s − 2·53-s − 4·65-s − 81-s − 4·85-s − 2·89-s + 4·97-s − 2·109-s − 2·113-s − 4·125-s + ⋯
L(s)  = 1  − 2·5-s + 2·13-s + 2·17-s + 3·25-s + 2·41-s − 2·49-s − 2·53-s − 4·65-s − 81-s − 4·85-s − 2·89-s + 4·97-s − 2·109-s − 2·113-s − 4·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.269389\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7994782846\)
\(L(\frac12)\) \(\approx\) \(0.7994782846\)
\(L(1)\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
11$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
97$C_1$ \( ( 1 - T )^{4} \)
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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

−10.55358473361831944388360473571, −9.926238049616057647830556242706, −9.284053314329414542714974887075, −9.224566675688286996073477659939, −8.360436181867569782726370961698, −8.274852459450495513823865956844, −7.87578858263733700707639570547, −7.65368702121397299826631055369, −7.02311306179865397577955313016, −6.66139748341650726099573964688, −5.91416898498638856922451051304, −5.87941412924884575262169694970, −4.98872780392329836583968713692, −4.63605909338792023553647171795, −3.95500902925516904691610965482, −3.74389105195115237189152840340, −3.13944010463524965719984642137, −2.94380580943645130112889393626, −1.53315551676988302622891983440, −0.971212245609352552474158287618, 0.971212245609352552474158287618, 1.53315551676988302622891983440, 2.94380580943645130112889393626, 3.13944010463524965719984642137, 3.74389105195115237189152840340, 3.95500902925516904691610965482, 4.63605909338792023553647171795, 4.98872780392329836583968713692, 5.87941412924884575262169694970, 5.91416898498638856922451051304, 6.66139748341650726099573964688, 7.02311306179865397577955313016, 7.65368702121397299826631055369, 7.87578858263733700707639570547, 8.274852459450495513823865956844, 8.360436181867569782726370961698, 9.224566675688286996073477659939, 9.284053314329414542714974887075, 9.926238049616057647830556242706, 10.55358473361831944388360473571

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