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Properties

Label 4-3072e2-1.1-c0e2-0-5
Degree $4$
Conductor $9437184$
Sign $1$
Analytic cond. $2.35048$
Root an. cond. $1.23819$
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  + 2·3-s + 3·9-s + 2·25-s + 4·27-s − 4·67-s + 4·75-s + 5·81-s + 2·121-s + 127-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 − 8·201-s + 211-s + 223-s + ⋯
L(s)  = 1  + 2·3-s + 3·9-s + 2·25-s + 4·27-s − 4·67-s + 4·75-s + 5·81-s + 2·121-s + 127-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 − 8·201-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9437184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9437184\)    =    \(2^{20} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(2.35048\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9437184,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.413127072\)
\(L(\frac12)\) \(\approx\) \(3.413127072\)
\(L(1)\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
good5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_2^2$ \( 1 + T^{4} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$ \( ( 1 + T )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2$ \( ( 1 + T^{2} )^{2} \)
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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

−8.881834692795028887061049922517, −8.794205864114964824314411705256, −8.399665349446361556498060224534, −8.072541743793536733190002331900, −7.52596335611397931786514473452, −7.37499643241552464779916070214, −7.01086211349315982832412088913, −6.62116947691019559362048009594, −6.14473742322609602884630383998, −5.72619460153615484409381773746, −4.92424057702025312292699280819, −4.74555622103127010106487657766, −4.39519548484192209994284929324, −3.85197022602547665222686457463, −3.33011194095108258170088571177, −3.14120914875801111181149470575, −2.54625549014459716940512483532, −2.30665643281194984544791808894, −1.44212126373452317595691176032, −1.20977330238521163189448484916, 1.20977330238521163189448484916, 1.44212126373452317595691176032, 2.30665643281194984544791808894, 2.54625549014459716940512483532, 3.14120914875801111181149470575, 3.33011194095108258170088571177, 3.85197022602547665222686457463, 4.39519548484192209994284929324, 4.74555622103127010106487657766, 4.92424057702025312292699280819, 5.72619460153615484409381773746, 6.14473742322609602884630383998, 6.62116947691019559362048009594, 7.01086211349315982832412088913, 7.37499643241552464779916070214, 7.52596335611397931786514473452, 8.072541743793536733190002331900, 8.399665349446361556498060224534, 8.794205864114964824314411705256, 8.881834692795028887061049922517

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