Properties

Label 4-768e2-1.1-c3e2-0-33
Degree $4$
Conductor $589824$
Sign $1$
Analytic cond. $2053.31$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·3-s + 73·9-s − 36·11-s + 250·25-s − 460·27-s + 360·33-s + 686·49-s − 1.69e3·59-s − 860·73-s − 2.50e3·75-s + 2.62e3·81-s − 2.70e3·83-s − 3.82e3·97-s − 2.62e3·99-s − 3.42e3·107-s − 1.69e3·121-s + 127-s + 131-s + 137-s + 139-s − 6.86e3·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4.39e3·169-s + ⋯
L(s)  = 1  − 1.92·3-s + 2.70·9-s − 0.986·11-s + 2·25-s − 3.27·27-s + 1.89·33-s + 2·49-s − 3.73·59-s − 1.37·73-s − 3.84·75-s + 3.60·81-s − 3.57·83-s − 3.99·97-s − 2.66·99-s − 3.08·107-s − 1.26·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 3.84·147-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 2·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(589824\)    =    \(2^{16} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(2053.31\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 589824,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + 10 T + p^{3} T^{2} \)
good5$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
7$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
11$C_2$ \( ( 1 + 18 T + p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 90 T + p^{3} T^{2} )( 1 + 90 T + p^{3} T^{2} ) \)
19$C_2$ \( ( 1 - 106 T + p^{3} T^{2} )( 1 + 106 T + p^{3} T^{2} ) \)
23$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 522 T + p^{3} T^{2} )( 1 + 522 T + p^{3} T^{2} ) \)
43$C_2$ \( ( 1 - 290 T + p^{3} T^{2} )( 1 + 290 T + p^{3} T^{2} ) \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 + 846 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 70 T + p^{3} T^{2} )( 1 + 70 T + p^{3} T^{2} ) \)
71$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 430 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + 1350 T + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 1026 T + p^{3} T^{2} )( 1 + 1026 T + p^{3} T^{2} ) \)
97$C_2$ \( ( 1 + 1910 T + p^{3} T^{2} )^{2} \)
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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

−9.677027439679656704433111633906, −9.651468943872047913663102183458, −8.797254785942251231226230309231, −8.551492298793469897848177120072, −7.72748620939740824652370208072, −7.50147024044170107059384577416, −6.90767182442380050739379272492, −6.72833020288962671913089458594, −6.01181207440862994914416260577, −5.77155422745996244257033902886, −5.16277232462089197402608662955, −4.99859750348933968220249928212, −4.25707746152116587579433104438, −4.08296670466566955581391606458, −2.92987602615562554795729466072, −2.64999616519168819934654621070, −1.35568208131777800669504827674, −1.26209032781018875984168653315, 0, 0, 1.26209032781018875984168653315, 1.35568208131777800669504827674, 2.64999616519168819934654621070, 2.92987602615562554795729466072, 4.08296670466566955581391606458, 4.25707746152116587579433104438, 4.99859750348933968220249928212, 5.16277232462089197402608662955, 5.77155422745996244257033902886, 6.01181207440862994914416260577, 6.72833020288962671913089458594, 6.90767182442380050739379272492, 7.50147024044170107059384577416, 7.72748620939740824652370208072, 8.551492298793469897848177120072, 8.797254785942251231226230309231, 9.651468943872047913663102183458, 9.677027439679656704433111633906

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