Properties

Label 4-483e2-1.1-c1e2-0-6
Degree $4$
Conductor $233289$
Sign $1$
Analytic cond. $14.8747$
Root an. cond. $1.96386$
Motivic weight $1$
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  − 3-s + 2·4-s + 5·7-s − 6·11-s − 2·12-s + 10·13-s + 6·17-s + 19-s − 5·21-s + 23-s + 5·25-s + 27-s + 10·28-s + 12·29-s − 5·31-s + 6·33-s + 7·37-s − 10·39-s − 2·43-s − 12·44-s − 6·47-s + 18·49-s − 6·51-s + 20·52-s − 12·53-s − 57-s + 6·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 4-s + 1.88·7-s − 1.80·11-s − 0.577·12-s + 2.77·13-s + 1.45·17-s + 0.229·19-s − 1.09·21-s + 0.208·23-s + 25-s + 0.192·27-s + 1.88·28-s + 2.22·29-s − 0.898·31-s + 1.04·33-s + 1.15·37-s − 1.60·39-s − 0.304·43-s − 1.80·44-s − 0.875·47-s + 18/7·49-s − 0.840·51-s + 2.77·52-s − 1.64·53-s − 0.132·57-s + 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(233289\)    =    \(3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(14.8747\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{483} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 233289,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.586471487\)
\(L(\frac12)\) \(\approx\) \(2.586471487\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
23$C_2$ \( 1 - T + T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−11.04874881047402161388100752147, −10.87699742890481836393325049772, −10.59552810844395674561022520822, −10.24188879175280374370204475737, −9.473331339187304138948343335740, −8.666904667437368768231839564594, −8.310238398719040413340336024257, −8.215957292124443851724440526804, −7.51641584421071846187434081491, −7.28709521874879857386142411812, −6.29181313736809311815566810704, −6.23549363334136228540821825558, −5.53684978854764559278433097917, −5.14138898611262881020016876981, −4.67814758332579250034171209939, −3.98305005149303116711957671665, −2.93954750076820013256659025927, −2.81032073272351323418807827555, −1.41228084486232279061352243294, −1.33050985624522978346679624229, 1.33050985624522978346679624229, 1.41228084486232279061352243294, 2.81032073272351323418807827555, 2.93954750076820013256659025927, 3.98305005149303116711957671665, 4.67814758332579250034171209939, 5.14138898611262881020016876981, 5.53684978854764559278433097917, 6.23549363334136228540821825558, 6.29181313736809311815566810704, 7.28709521874879857386142411812, 7.51641584421071846187434081491, 8.215957292124443851724440526804, 8.310238398719040413340336024257, 8.666904667437368768231839564594, 9.473331339187304138948343335740, 10.24188879175280374370204475737, 10.59552810844395674561022520822, 10.87699742890481836393325049772, 11.04874881047402161388100752147

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