Properties

Label 4-1891125-1.1-c1e2-0-6
Degree $4$
Conductor $1891125$
Sign $-1$
Analytic cond. $120.579$
Root an. cond. $3.31374$
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  + 2·3-s − 3·4-s + 5-s + 3·9-s − 6·12-s + 4·13-s + 2·15-s + 5·16-s − 4·17-s − 3·20-s + 25-s + 4·27-s − 9·36-s + 8·39-s + 10·41-s + 3·45-s − 16·47-s + 10·48-s − 14·49-s − 8·51-s − 12·52-s + 20·53-s − 8·59-s − 6·60-s − 4·61-s − 3·64-s + 4·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s + 0.447·5-s + 9-s − 1.73·12-s + 1.10·13-s + 0.516·15-s + 5/4·16-s − 0.970·17-s − 0.670·20-s + 1/5·25-s + 0.769·27-s − 3/2·36-s + 1.28·39-s + 1.56·41-s + 0.447·45-s − 2.33·47-s + 1.44·48-s − 2·49-s − 1.12·51-s − 1.66·52-s + 2.74·53-s − 1.04·59-s − 0.774·60-s − 0.512·61-s − 3/8·64-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1891125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1891125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1891125\)    =    \(3^{2} \cdot 5^{3} \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(120.579\)
Root analytic conductor: \(3.31374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1891125} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1891125,\ (\ :1/2, 1/2),\ -1)\)

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

−7.68003844385765985083943135134, −7.37759566583217556708649005666, −6.61508786477142818172091419880, −6.35659216039682194641838348042, −5.86497653842965752491328535183, −5.31090056485434020639114264233, −4.70227288054663401602265605108, −4.52160444884874669934496061646, −3.96065425433574684505069552401, −3.58121207448309608095492288703, −3.01549784140686353642765162463, −2.50062601974282194950611376290, −1.68625435843919962684270563812, −1.18674782867264970790488068509, 0, 1.18674782867264970790488068509, 1.68625435843919962684270563812, 2.50062601974282194950611376290, 3.01549784140686353642765162463, 3.58121207448309608095492288703, 3.96065425433574684505069552401, 4.52160444884874669934496061646, 4.70227288054663401602265605108, 5.31090056485434020639114264233, 5.86497653842965752491328535183, 6.35659216039682194641838348042, 6.61508786477142818172091419880, 7.37759566583217556708649005666, 7.68003844385765985083943135134

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