Properties

Label 4-84e2-1.1-c1e2-0-4
Degree $4$
Conductor $7056$
Sign $-1$
Analytic cond. $0.449896$
Root an. cond. $0.818989$
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-s − 4-s − 4·5-s + 3·8-s + 9-s + 4·10-s − 4·13-s − 16-s − 12·17-s − 18-s + 4·20-s + 2·25-s + 4·26-s − 4·29-s − 5·32-s + 12·34-s − 36-s + 12·37-s − 12·40-s + 4·41-s − 4·45-s + 49-s − 2·50-s + 4·52-s + 12·53-s + 4·58-s − 4·61-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s + 1/3·9-s + 1.26·10-s − 1.10·13-s − 1/4·16-s − 2.91·17-s − 0.235·18-s + 0.894·20-s + 2/5·25-s + 0.784·26-s − 0.742·29-s − 0.883·32-s + 2.05·34-s − 1/6·36-s + 1.97·37-s − 1.89·40-s + 0.624·41-s − 0.596·45-s + 1/7·49-s − 0.282·50-s + 0.554·52-s + 1.64·53-s + 0.525·58-s − 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(0.449896\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 7056,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−11.55595081754559805664353277943, −11.04847280565543144871395360807, −10.51953132809604428367636932730, −9.724661210726173973340366981146, −9.198895146278336601365818133221, −8.672365196683144779961098572501, −8.068098981215555715874277637796, −7.38325958034671995701748313853, −7.25047783802838427330028450541, −6.19643457342138342025418009798, −4.94701466488767178340177905762, −4.13981751462080886088964913424, −4.13559084050773741974089362056, −2.39234840548789475685588947322, 0, 2.39234840548789475685588947322, 4.13559084050773741974089362056, 4.13981751462080886088964913424, 4.94701466488767178340177905762, 6.19643457342138342025418009798, 7.25047783802838427330028450541, 7.38325958034671995701748313853, 8.068098981215555715874277637796, 8.672365196683144779961098572501, 9.198895146278336601365818133221, 9.724661210726173973340366981146, 10.51953132809604428367636932730, 11.04847280565543144871395360807, 11.55595081754559805664353277943

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