Properties

Label 4-2736e2-1.1-c0e2-0-2
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $1.86443$
Root an. cond. $1.16852$
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  − 2·7-s + 2·19-s + 25-s + 2·43-s + 49-s + 2·61-s − 2·73-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 2·7-s + 2·19-s + 25-s + 2·43-s + 49-s + 2·61-s − 2·73-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1.86443\)
Root analytic conductor: \(1.16852\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.029378454\)
\(L(\frac12)\) \(\approx\) \(1.029378454\)
\(L(1)\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_2$ \( ( 1 + T + T^{2} )^{2} \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2^2$ \( 1 - T^{2} + T^{4} \)
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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
47$C_2^2$ \( 1 - T^{2} + T^{4} \)
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$C_2$ \( ( 1 - T + T^{2} )^{2} \)
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$C_2$ \( ( 1 + T + T^{2} )^{2} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−9.235990735237979540840943919229, −8.900703449638181422581573172415, −8.619410630756537695021763134250, −7.961290920602328844846940300758, −7.63189394962964192311821693812, −7.15794941416174872333860017999, −7.04953657515592276335939624692, −6.46591072558838029086479624716, −6.23514015606536678206407764949, −5.79248924258483129659513496169, −5.26621118161322420950977046831, −5.16862967248409659090433585657, −4.27314550724798512049580917546, −4.10286384222054149602629525690, −3.29625699884825631953904319194, −3.25098285817593569644123243872, −2.78442535581043261675241404688, −2.27302423973733520947508430752, −1.33300774873847751313801934917, −0.70888517576267816430448333051, 0.70888517576267816430448333051, 1.33300774873847751313801934917, 2.27302423973733520947508430752, 2.78442535581043261675241404688, 3.25098285817593569644123243872, 3.29625699884825631953904319194, 4.10286384222054149602629525690, 4.27314550724798512049580917546, 5.16862967248409659090433585657, 5.26621118161322420950977046831, 5.79248924258483129659513496169, 6.23514015606536678206407764949, 6.46591072558838029086479624716, 7.04953657515592276335939624692, 7.15794941416174872333860017999, 7.63189394962964192311821693812, 7.961290920602328844846940300758, 8.619410630756537695021763134250, 8.900703449638181422581573172415, 9.235990735237979540840943919229

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