Properties

Label 4-21168-1.1-c1e2-0-10
Degree $4$
Conductor $21168$
Sign $-1$
Analytic cond. $1.34969$
Root an. cond. $1.07785$
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 − 3-s − 4-s − 6-s − 3·8-s + 9-s − 8·11-s + 12-s − 4·13-s − 16-s + 18-s − 8·22-s + 3·24-s − 6·25-s − 4·26-s − 27-s + 5·32-s + 8·33-s − 36-s + 12·37-s + 4·39-s + 8·44-s + 48-s + 49-s − 6·50-s + 4·52-s − 54-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 2.41·11-s + 0.288·12-s − 1.10·13-s − 1/4·16-s + 0.235·18-s − 1.70·22-s + 0.612·24-s − 6/5·25-s − 0.784·26-s − 0.192·27-s + 0.883·32-s + 1.39·33-s − 1/6·36-s + 1.97·37-s + 0.640·39-s + 1.20·44-s + 0.144·48-s + 1/7·49-s − 0.848·50-s + 0.554·52-s − 0.136·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21168\)    =    \(2^{4} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1.34969\)
Root analytic conductor: \(1.07785\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 21168,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−10.51953132809604428367636932730, −10.18331996404873608095625884352, −9.457705101580370517739336150901, −9.198895146278336601365818133221, −8.068098981215555715874277637796, −7.81295430367747747098376616892, −7.38325958034671995701748313853, −6.19643457342138342025418009798, −5.94607665445287604157300008429, −5.02709416296405066539370067618, −4.94701466488767178340177905762, −4.13981751462080886088964913424, −3.05422074105458389777226041971, −2.39234840548789475685588947322, 0, 2.39234840548789475685588947322, 3.05422074105458389777226041971, 4.13981751462080886088964913424, 4.94701466488767178340177905762, 5.02709416296405066539370067618, 5.94607665445287604157300008429, 6.19643457342138342025418009798, 7.38325958034671995701748313853, 7.81295430367747747098376616892, 8.068098981215555715874277637796, 9.198895146278336601365818133221, 9.457705101580370517739336150901, 10.18331996404873608095625884352, 10.51953132809604428367636932730

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