Properties

Label 4-33e4-1.1-c2e2-0-0
Degree $4$
Conductor $1185921$
Sign $1$
Analytic cond. $880.492$
Root an. cond. $5.44730$
Motivic weight $2$
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  + 6·4-s − 2·7-s − 32·13-s + 20·16-s − 66·19-s + 18·25-s − 12·28-s + 62·31-s − 114·37-s − 95·49-s − 192·52-s − 210·61-s + 24·64-s − 206·67-s + 94·73-s − 396·76-s − 46·79-s + 64·91-s − 50·97-s + 108·100-s − 50·103-s + 238·109-s − 40·112-s + 372·124-s + 127-s + 131-s + 132·133-s + ⋯
L(s)  = 1  + 3/2·4-s − 2/7·7-s − 2.46·13-s + 5/4·16-s − 3.47·19-s + 0.719·25-s − 3/7·28-s + 2·31-s − 3.08·37-s − 1.93·49-s − 3.69·52-s − 3.44·61-s + 3/8·64-s − 3.07·67-s + 1.28·73-s − 5.21·76-s − 0.582·79-s + 0.703·91-s − 0.515·97-s + 1.07·100-s − 0.485·103-s + 2.18·109-s − 0.357·112-s + 3·124-s + 0.00787·127-s + 0.00763·131-s + 0.992·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(880.492\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1185921,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3106115801\)
\(L(\frac12)\) \(\approx\) \(0.3106115801\)
\(L(2)\) 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)$
bad3 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( 1 - 3 p T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 18 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 450 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 33 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 480 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1520 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 57 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3120 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 720 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 + 2320 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 2160 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 105 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 103 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 4030 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 47 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 23 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 2528 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 13250 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
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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

−10.28555928844182302158162980085, −9.484760928183775008286914215954, −9.085985262835343717118712015008, −8.521736553846322655351591783424, −8.269620517384439595298854174812, −7.71163392046260910560159710946, −7.27264667033755079949860006464, −6.82678411706001966354791793641, −6.70164544711601184623792006964, −6.12963317728884603205604906866, −5.94020861529580906403635350847, −4.97072970831393252833162455447, −4.59064631179632840893732752467, −4.50078700777250284682363521745, −3.44245055068357922361408638735, −2.86935628219156447016442233487, −2.62946867691116239107675466686, −1.81960435233667253479874611678, −1.77430814633204411547754937482, −0.14728230772656822312402314515, 0.14728230772656822312402314515, 1.77430814633204411547754937482, 1.81960435233667253479874611678, 2.62946867691116239107675466686, 2.86935628219156447016442233487, 3.44245055068357922361408638735, 4.50078700777250284682363521745, 4.59064631179632840893732752467, 4.97072970831393252833162455447, 5.94020861529580906403635350847, 6.12963317728884603205604906866, 6.70164544711601184623792006964, 6.82678411706001966354791793641, 7.27264667033755079949860006464, 7.71163392046260910560159710946, 8.269620517384439595298854174812, 8.521736553846322655351591783424, 9.085985262835343717118712015008, 9.484760928183775008286914215954, 10.28555928844182302158162980085

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