Properties

Label 4-39e2-1.1-c18e2-0-1
Degree $4$
Conductor $1521$
Sign $1$
Analytic cond. $6416.10$
Root an. cond. $8.94989$
Motivic weight $18$
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  + 3.93e4·3-s − 5.24e5·4-s + 1.16e9·9-s − 2.06e10·12-s + 7.19e9·13-s + 2.06e11·16-s − 7.62e12·25-s + 3.05e13·27-s − 6.09e14·36-s + 2.83e14·39-s − 1.46e15·43-s + 8.11e15·48-s + 2.75e15·49-s − 3.77e15·52-s + 1.89e16·61-s − 7.20e16·64-s − 3.00e17·75-s − 2.81e17·79-s + 7.50e17·81-s + 4.00e18·100-s − 5.21e18·103-s − 1.59e19·108-s + 8.36e18·117-s − 1.11e19·121-s + 127-s − 5.75e19·129-s + 131-s + ⋯
L(s)  = 1  + 2·3-s − 2·4-s + 3·9-s − 4·12-s + 0.678·13-s + 3·16-s − 2·25-s + 4·27-s − 6·36-s + 1.35·39-s − 2.90·43-s + 6·48-s + 1.69·49-s − 1.35·52-s + 1.62·61-s − 4·64-s − 4·75-s − 2.34·79-s + 5·81-s + 4·100-s − 3.99·103-s − 8·108-s + 2.03·117-s − 2·121-s − 5.81·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+9)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(6416.10\)
Root analytic conductor: \(8.94989\)
Motivic weight: \(18\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1521,\ (\ :9, 9),\ 1)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(2.952865025\)
\(L(\frac12)\) \(\approx\) \(2.952865025\)
\(L(10)\) 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$C_1$ \( ( 1 - p^{9} T )^{2} \)
13$C_2$ \( 1 - 7197541846 T + p^{18} T^{2} \)
good2$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 77549186 T + p^{18} T^{2} )( 1 + 77549186 T + p^{18} T^{2} ) \)
11$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - p^{9} T )^{2}( 1 + p^{9} T )^{2} \)
19$C_2$ \( ( 1 - 308559680858 T + p^{18} T^{2} )( 1 + 308559680858 T + p^{18} T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - p^{9} T )^{2}( 1 + p^{9} T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{9} T )^{2}( 1 + p^{9} T )^{2} \)
31$C_2$ \( ( 1 - 50018992173358 T + p^{18} T^{2} )( 1 + 50018992173358 T + p^{18} T^{2} ) \)
37$C_2$ \( ( 1 - 23240947030054 T + p^{18} T^{2} )( 1 + 23240947030054 T + p^{18} T^{2} ) \)
41$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 730385642547286 T + p^{18} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - p^{9} T )^{2}( 1 + p^{9} T )^{2} \)
59$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 9487161099916918 T + p^{18} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 41747295001607494 T + p^{18} T^{2} )( 1 + 41747295001607494 T + p^{18} T^{2} ) \)
71$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 29908998244279726 T + p^{18} T^{2} )( 1 + 29908998244279726 T + p^{18} T^{2} ) \)
79$C_2$ \( ( 1 + 140655567501204338 T + p^{18} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{18} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 140873967896062466 T + p^{18} T^{2} )( 1 + 140873967896062466 T + p^{18} T^{2} ) \)
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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

−13.15298627470623359341446478666, −12.44333287379212321129771001903, −11.72438028062121135979584031833, −10.43675031828537215297964601643, −9.960000200394304511503400107175, −9.607674717835733369238417731210, −9.029895052131635350987039983304, −8.354056994453547721184981004849, −8.321663605523985923545252663966, −7.60801394350844151101269361171, −6.80216166738754828287703267536, −5.70310285243492338971731032160, −5.02064758495337757302927168666, −4.14505679276865527834955683370, −3.94809117774332293775672768667, −3.41270226914847917844602319010, −2.66863480659590350305211206742, −1.69210466534197031194707738663, −1.26685120018441289932379391096, −0.37405293840099820245824893154, 0.37405293840099820245824893154, 1.26685120018441289932379391096, 1.69210466534197031194707738663, 2.66863480659590350305211206742, 3.41270226914847917844602319010, 3.94809117774332293775672768667, 4.14505679276865527834955683370, 5.02064758495337757302927168666, 5.70310285243492338971731032160, 6.80216166738754828287703267536, 7.60801394350844151101269361171, 8.321663605523985923545252663966, 8.354056994453547721184981004849, 9.029895052131635350987039983304, 9.607674717835733369238417731210, 9.960000200394304511503400107175, 10.43675031828537215297964601643, 11.72438028062121135979584031833, 12.44333287379212321129771001903, 13.15298627470623359341446478666

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