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Properties

Label 4-42e4-1.1-c0e2-0-1
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $0.775017$
Root an. cond. $0.938270$
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 + 3·4-s − 4·8-s + 5·16-s + 4·29-s − 6·32-s − 8·58-s + 7·64-s + 12·116-s + 2·121-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 4·29-s − 6·32-s − 8·58-s + 7·64-s + 12·116-s + 2·121-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(0.775017\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5141235334\)
\(L(\frac12)\) \(\approx\) \(0.5141235334\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$ \( ( 1 - T )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_2^2$ \( 1 + T^{4} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2^2$ \( 1 + T^{4} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.573851337685422654283782445352, −9.399370977327345979084862035326, −8.756805946660462987181752486849, −8.499641658719668833265027931500, −8.275388977725322122955113335567, −7.937278212016017953230038094861, −7.39285223842948294178162293358, −6.97573572784268382389450002190, −6.68520672525392673578417997868, −6.35924189427171884729468779059, −5.85993416270745834999731613801, −5.50631582440132158687708465641, −4.66648982491208405075658801846, −4.47532135661802338154032159259, −3.37700479978420989531265258823, −3.21801443839988885241828178761, −2.51730789116789760703552203771, −2.21046504766036396850092245880, −1.32082409731483583590983312601, −0.840214520965958584451282685107, 0.840214520965958584451282685107, 1.32082409731483583590983312601, 2.21046504766036396850092245880, 2.51730789116789760703552203771, 3.21801443839988885241828178761, 3.37700479978420989531265258823, 4.47532135661802338154032159259, 4.66648982491208405075658801846, 5.50631582440132158687708465641, 5.85993416270745834999731613801, 6.35924189427171884729468779059, 6.68520672525392673578417997868, 6.97573572784268382389450002190, 7.39285223842948294178162293358, 7.937278212016017953230038094861, 8.275388977725322122955113335567, 8.499641658719668833265027931500, 8.756805946660462987181752486849, 9.399370977327345979084862035326, 9.573851337685422654283782445352

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