Properties

Label 4-165375-1.1-c1e2-0-9
Degree $4$
Conductor $165375$
Sign $1$
Analytic cond. $10.5444$
Root an. cond. $1.80200$
Motivic weight $1$
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  + 2·2-s + 3-s − 4-s + 5-s + 2·6-s − 8·8-s + 9-s + 2·10-s − 12-s + 4·13-s + 15-s − 7·16-s + 2·18-s − 20-s − 8·24-s + 25-s + 8·26-s + 27-s + 2·30-s + 14·32-s − 36-s + 4·39-s − 8·40-s + 20·41-s + 45-s − 7·48-s − 7·49-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 2.82·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s + 1.10·13-s + 0.258·15-s − 7/4·16-s + 0.471·18-s − 0.223·20-s − 1.63·24-s + 1/5·25-s + 1.56·26-s + 0.192·27-s + 0.365·30-s + 2.47·32-s − 1/6·36-s + 0.640·39-s − 1.26·40-s + 3.12·41-s + 0.149·45-s − 1.01·48-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165375 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165375 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(165375\)    =    \(3^{3} \cdot 5^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(10.5444\)
Root analytic conductor: \(1.80200\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{165375} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 165375,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.063059717\)
\(L(\frac12)\) \(\approx\) \(3.063059717\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_2$ \( 1 + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−8.890589428908360265391160029852, −8.843956595114680827452383410006, −8.627000477337622059216224790867, −7.68003844385765985083943135134, −7.37485778338999061621083536758, −6.39356342307905775022074491331, −5.91397914498304472267826736485, −5.86497653842965752491328535183, −5.03119182641673168471146566272, −4.52160444884874669934496061646, −4.09922216243454775532088496022, −3.56571005686312005410318267112, −3.01549784140686353642765162463, −2.32940333880865099555943992805, −0.964820104627525128067217267144, 0.964820104627525128067217267144, 2.32940333880865099555943992805, 3.01549784140686353642765162463, 3.56571005686312005410318267112, 4.09922216243454775532088496022, 4.52160444884874669934496061646, 5.03119182641673168471146566272, 5.86497653842965752491328535183, 5.91397914498304472267826736485, 6.39356342307905775022074491331, 7.37485778338999061621083536758, 7.68003844385765985083943135134, 8.627000477337622059216224790867, 8.843956595114680827452383410006, 8.890589428908360265391160029852

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