Properties

Label 4-882e2-1.1-c2e2-0-1
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $577.573$
Root an. cond. $4.90232$
Motivic weight $2$
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  − 2·4-s − 16·13-s + 4·16-s + 32·19-s + 32·25-s − 88·31-s − 68·37-s − 80·43-s + 32·52-s − 100·61-s − 8·64-s + 16·67-s + 32·73-s − 64·76-s − 152·79-s − 352·97-s − 64·100-s + 56·103-s − 46·121-s + 176·124-s + 127-s + 131-s + 137-s + 139-s + 136·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.23·13-s + 1/4·16-s + 1.68·19-s + 1.27·25-s − 2.83·31-s − 1.83·37-s − 1.86·43-s + 8/13·52-s − 1.63·61-s − 1/8·64-s + 0.238·67-s + 0.438·73-s − 0.842·76-s − 1.92·79-s − 3.62·97-s − 0.639·100-s + 0.543·103-s − 0.380·121-s + 1.41·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.918·148-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(577.573\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7537953800\)
\(L(\frac12)\) \(\approx\) \(0.7537953800\)
\(L(2)\) 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)$
bad2$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 32 T^{2} + p^{4} T^{4} \)
11$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 416 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 770 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1664 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 44 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 34 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 1184 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 2782 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4160 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 5810 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7490 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 76 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 334 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 15680 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 176 T + p^{2} T^{2} )^{2} \)
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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

−10.30043516936086576043929229061, −9.615085954447740487433903839774, −9.406925528344981426811008520087, −8.784479815182659277731965580058, −8.733354996297716235976672692723, −7.999945832806824503288140739230, −7.50619857545755213365572744010, −7.19682686579861573137408694641, −6.92627384444924372589096723994, −6.30597137565107976123829190076, −5.49657973238433766426055376084, −5.21975034065573553757766612376, −5.10652681356493178402301024112, −4.34906018890944444377569569981, −3.77125762750569263611071986298, −3.15559971172624801595723374441, −2.91145975929704736032018073339, −1.85059320603374405960128428960, −1.42577528943012396507964750190, −0.28912113765826791572762518793, 0.28912113765826791572762518793, 1.42577528943012396507964750190, 1.85059320603374405960128428960, 2.91145975929704736032018073339, 3.15559971172624801595723374441, 3.77125762750569263611071986298, 4.34906018890944444377569569981, 5.10652681356493178402301024112, 5.21975034065573553757766612376, 5.49657973238433766426055376084, 6.30597137565107976123829190076, 6.92627384444924372589096723994, 7.19682686579861573137408694641, 7.50619857545755213365572744010, 7.999945832806824503288140739230, 8.733354996297716235976672692723, 8.784479815182659277731965580058, 9.406925528344981426811008520087, 9.615085954447740487433903839774, 10.30043516936086576043929229061

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