Properties

Label 4-408375-1.1-c1e2-0-2
Degree $4$
Conductor $408375$
Sign $-1$
Analytic cond. $26.0383$
Root an. cond. $2.25893$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·4-s − 5-s + 9-s − 4·11-s − 3·12-s + 4·13-s − 15-s + 5·16-s + 3·20-s + 25-s + 27-s − 4·29-s − 4·33-s − 3·36-s + 4·39-s + 20·41-s − 8·43-s + 12·44-s − 45-s − 16·47-s + 5·48-s − 14·49-s − 12·52-s + 20·53-s + 4·55-s + 3·60-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.866·12-s + 1.10·13-s − 0.258·15-s + 5/4·16-s + 0.670·20-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 0.696·33-s − 1/2·36-s + 0.640·39-s + 3.12·41-s − 1.21·43-s + 1.80·44-s − 0.149·45-s − 2.33·47-s + 0.721·48-s − 2·49-s − 1.66·52-s + 2.74·53-s + 0.539·55-s + 0.387·60-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 408375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 408375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(408375\)    =    \(3^{3} \cdot 5^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(26.0383\)
Root analytic conductor: \(2.25893\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 408375,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_1$ \( 1 - T \)
5$C_1$ \( 1 + T \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.327185665547056415560546606477, −8.168912727661556984599726575199, −7.68003844385765985083943135134, −7.22896894442775896645972178642, −6.58372794841850391610973143735, −5.86497653842965752491328535183, −5.57792290694465802264005507804, −4.87349408210187854122798421456, −4.52160444884874669934496061646, −4.01446647923640082173818964259, −3.45000960347210639177784403115, −3.01549784140686353642765162463, −2.13143823519478900872906310831, −1.11050393637334551379963274590, 0, 1.11050393637334551379963274590, 2.13143823519478900872906310831, 3.01549784140686353642765162463, 3.45000960347210639177784403115, 4.01446647923640082173818964259, 4.52160444884874669934496061646, 4.87349408210187854122798421456, 5.57792290694465802264005507804, 5.86497653842965752491328535183, 6.58372794841850391610973143735, 7.22896894442775896645972178642, 7.68003844385765985083943135134, 8.168912727661556984599726575199, 8.327185665547056415560546606477

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