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Properties

Label 4-1400e2-1.1-c0e2-0-3
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $0.488169$
Root an. cond. $0.835877$
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  − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 4·14-s + 5·16-s + 6·28-s − 6·32-s + 3·49-s − 8·56-s + 7·64-s − 81-s − 6·98-s + 10·112-s + 4·113-s + 2·121-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 4·14-s + 5·16-s + 6·28-s − 6·32-s + 3·49-s − 8·56-s + 7·64-s − 81-s − 6·98-s + 10·112-s + 4·113-s + 2·121-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.488169\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6027764964\)
\(L(\frac12)\) \(\approx\) \(0.6027764964\)
\(L(1)\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.00145615138226631232542030836, −9.539879422217702921626735452615, −8.978059875023467949933676766683, −8.724630634173229318593966903615, −8.375555927139315546491184104959, −8.140939753357669301451261887400, −7.60278033682101438168505721462, −7.29874567383983547375946247741, −7.11327451894400425567814646014, −6.41993972291292251686613973530, −5.89255768538943068111472731722, −5.67631954318862370430598278022, −4.95822600815069049158835218240, −4.61969348407239127380924462246, −3.82044374204711435348358272558, −3.26516048459389391352190780139, −2.49304226006731570545255365727, −2.11881067759400590285768240661, −1.53885203955073937888686704228, −0.979259789588622952788754477622, 0.979259789588622952788754477622, 1.53885203955073937888686704228, 2.11881067759400590285768240661, 2.49304226006731570545255365727, 3.26516048459389391352190780139, 3.82044374204711435348358272558, 4.61969348407239127380924462246, 4.95822600815069049158835218240, 5.67631954318862370430598278022, 5.89255768538943068111472731722, 6.41993972291292251686613973530, 7.11327451894400425567814646014, 7.29874567383983547375946247741, 7.60278033682101438168505721462, 8.140939753357669301451261887400, 8.375555927139315546491184104959, 8.724630634173229318593966903615, 8.978059875023467949933676766683, 9.539879422217702921626735452615, 10.00145615138226631232542030836

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