Properties

Label 4-390e2-1.1-c1e2-0-19
Degree $4$
Conductor $152100$
Sign $-1$
Analytic cond. $9.69802$
Root an. cond. $1.76469$
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 + 4-s − 8·7-s + 9-s + 2·12-s − 2·13-s + 16-s + 12·19-s − 16·21-s + 25-s − 4·27-s − 8·28-s − 12·31-s + 36-s − 4·37-s − 4·39-s − 20·43-s + 2·48-s + 34·49-s − 2·52-s + 24·57-s + 4·61-s − 8·63-s + 64-s − 24·67-s + 20·73-s + 2·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 3.02·7-s + 1/3·9-s + 0.577·12-s − 0.554·13-s + 1/4·16-s + 2.75·19-s − 3.49·21-s + 1/5·25-s − 0.769·27-s − 1.51·28-s − 2.15·31-s + 1/6·36-s − 0.657·37-s − 0.640·39-s − 3.04·43-s + 0.288·48-s + 34/7·49-s − 0.277·52-s + 3.17·57-s + 0.512·61-s − 1.00·63-s + 1/8·64-s − 2.93·67-s + 2.34·73-s + 0.230·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(9.69802\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{152100} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 152100,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−9.161137327460865013114506016469, −8.868774471216060607456403615280, −7.83531743824800124245932651769, −7.66670900603824463060194393110, −6.87551315707654393130092819540, −6.87025308528864829107350186696, −6.22340654796584066361422256816, −5.34711726354022861021902847980, −5.33976892050247260934145252130, −3.81800601968201289194081442487, −3.53009310754873249134401547682, −2.96623355538274298367536210534, −2.88253971705917726242541361642, −1.69050510807477611643559790122, 0, 1.69050510807477611643559790122, 2.88253971705917726242541361642, 2.96623355538274298367536210534, 3.53009310754873249134401547682, 3.81800601968201289194081442487, 5.33976892050247260934145252130, 5.34711726354022861021902847980, 6.22340654796584066361422256816, 6.87025308528864829107350186696, 6.87551315707654393130092819540, 7.66670900603824463060194393110, 7.83531743824800124245932651769, 8.868774471216060607456403615280, 9.161137327460865013114506016469

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