Properties

Label 4-87362-1.1-c1e2-0-2
Degree $4$
Conductor $87362$
Sign $-1$
Analytic cond. $5.57027$
Root an. cond. $1.53627$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 5-s − 8-s − 3·9-s − 10-s − 11-s + 6·13-s + 3·16-s − 3·18-s − 6·19-s + 20-s − 22-s − 8·23-s − 7·25-s + 6·26-s − 6·29-s + 3·32-s + 3·36-s − 6·38-s + 40-s + 44-s + 3·45-s − 8·46-s + 14·47-s − 5·49-s − 7·50-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.353·8-s − 9-s − 0.316·10-s − 0.301·11-s + 1.66·13-s + 3/4·16-s − 0.707·18-s − 1.37·19-s + 0.223·20-s − 0.213·22-s − 1.66·23-s − 7/5·25-s + 1.17·26-s − 1.11·29-s + 0.530·32-s + 1/2·36-s − 0.973·38-s + 0.158·40-s + 0.150·44-s + 0.447·45-s − 1.17·46-s + 2.04·47-s − 5/7·49-s − 0.989·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87362 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(87362\)    =    \(2 \cdot 11^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(5.57027\)
Root analytic conductor: \(1.53627\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 87362,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T^{2} ) \)
11$C_2$ \( 1 + T + p T^{2} \)
19$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 71 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 67 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.254706501854957364501766642291, −8.907919440505075761265878776745, −8.309340271395688103631663104998, −8.042888692826423933797120299928, −7.62081938828138297229725147470, −6.69674548376858752943068105991, −6.03163169286647759642201268611, −5.77380920252080213203721246978, −5.37229546840605511929667692095, −4.29062859807934539822577422920, −4.03219376486958036593598966702, −3.64516090541421099591304301480, −2.70540793641000470403168683387, −1.72119349768487735074454778353, 0, 1.72119349768487735074454778353, 2.70540793641000470403168683387, 3.64516090541421099591304301480, 4.03219376486958036593598966702, 4.29062859807934539822577422920, 5.37229546840605511929667692095, 5.77380920252080213203721246978, 6.03163169286647759642201268611, 6.69674548376858752943068105991, 7.62081938828138297229725147470, 8.042888692826423933797120299928, 8.309340271395688103631663104998, 8.907919440505075761265878776745, 9.254706501854957364501766642291

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