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Properties

Label 4-3528e2-1.1-c0e2-0-14
Degree $4$
Conductor $12446784$
Sign $1$
Analytic cond. $3.10006$
Root an. cond. $1.32691$
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 + 4·8-s + 5·16-s + 2·25-s + 6·32-s + 4·50-s + 7·64-s − 4·67-s + 6·100-s − 4·107-s − 2·121-s + 127-s + 8·128-s + 131-s − 8·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 5·16-s + 2·25-s + 6·32-s + 4·50-s + 7·64-s − 4·67-s + 6·100-s − 4·107-s − 2·121-s + 127-s + 8·128-s + 131-s − 8·134-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12446784\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.10006\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12446784,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.549748558\)
\(L(\frac12)\) \(\approx\) \(6.549748558\)
\(L(1)\) 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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_2^2$ \( 1 + T^{4} \)
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_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2^2$ \( 1 + T^{4} \)
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$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$ \( ( 1 + T )^{4} \)
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$C_2^2$ \( 1 + T^{4} \)
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

−8.774029967312598213741477833416, −8.531196624933967136851025177859, −7.949169812962075299915951881535, −7.71170454377112592635782080580, −7.19305344069580450103399159983, −7.02850629375455020609382033941, −6.48288273858280370695083747989, −6.39339894044203444175283931800, −5.71632295172605145089621620672, −5.62007987239258451523621769986, −4.98794655241667327086246712655, −4.83019438946390675348486425956, −4.27995003549594251522400254868, −4.11743269429401872318814618718, −3.34453522607252592265456941674, −3.22730490348649438914891577355, −2.54020018782380841957006951260, −2.47683963146660941994138910240, −1.38101887634597907693073166957, −1.36467350816363020245700921873, 1.36467350816363020245700921873, 1.38101887634597907693073166957, 2.47683963146660941994138910240, 2.54020018782380841957006951260, 3.22730490348649438914891577355, 3.34453522607252592265456941674, 4.11743269429401872318814618718, 4.27995003549594251522400254868, 4.83019438946390675348486425956, 4.98794655241667327086246712655, 5.62007987239258451523621769986, 5.71632295172605145089621620672, 6.39339894044203444175283931800, 6.48288273858280370695083747989, 7.02850629375455020609382033941, 7.19305344069580450103399159983, 7.71170454377112592635782080580, 7.949169812962075299915951881535, 8.531196624933967136851025177859, 8.774029967312598213741477833416

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