Properties

Label 4-540e2-1.1-c0e2-0-0
Degree $4$
Conductor $291600$
Sign $1$
Analytic cond. $0.0726276$
Root an. cond. $0.519129$
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-s + 5-s − 7-s + 8-s − 10-s + 14-s − 16-s + 23-s − 29-s − 35-s + 40-s − 41-s + 2·43-s − 46-s + 47-s + 49-s − 56-s + 58-s + 61-s + 64-s − 67-s + 70-s − 80-s + 82-s + 83-s − 2·86-s + 2·89-s + ⋯
L(s)  = 1  − 2-s + 5-s − 7-s + 8-s − 10-s + 14-s − 16-s + 23-s − 29-s − 35-s + 40-s − 41-s + 2·43-s − 46-s + 47-s + 49-s − 56-s + 58-s + 61-s + 64-s − 67-s + 70-s − 80-s + 82-s + 83-s − 2·86-s + 2·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(291600\)    =    \(2^{4} \cdot 3^{6} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.0726276\)
Root analytic conductor: \(0.519129\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 291600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4608339515\)
\(L(\frac12)\) \(\approx\) \(0.4608339515\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
29$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
43$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
89$C_2$ \( ( 1 - T + T^{2} )^{2} \)
97$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
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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

−10.97017062096837622727523026908, −10.62593273313949135147153047916, −10.27620481528115088537215819704, −9.872053017151147102207854555176, −9.329979014909354021152153654827, −9.232752757954356762436883107694, −8.832143153405060182665829442173, −8.319425501913797546385383074319, −7.52519517997436693382247180723, −7.41238559431710544597151247096, −6.83451333307522758265591328945, −6.20544926634159153008711595560, −5.89319377979588682701216326736, −5.24649080765084844197287196045, −4.79864218464574690665485036118, −3.96557977926722032178687934316, −3.52664835806404417048596800116, −2.59372777403914185558863819939, −2.07320807731151890144516204983, −1.06818385127464349816642610461, 1.06818385127464349816642610461, 2.07320807731151890144516204983, 2.59372777403914185558863819939, 3.52664835806404417048596800116, 3.96557977926722032178687934316, 4.79864218464574690665485036118, 5.24649080765084844197287196045, 5.89319377979588682701216326736, 6.20544926634159153008711595560, 6.83451333307522758265591328945, 7.41238559431710544597151247096, 7.52519517997436693382247180723, 8.319425501913797546385383074319, 8.832143153405060182665829442173, 9.232752757954356762436883107694, 9.329979014909354021152153654827, 9.872053017151147102207854555176, 10.27620481528115088537215819704, 10.62593273313949135147153047916, 10.97017062096837622727523026908

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