Properties

Label 4-12672-1.1-c1e2-0-0
Degree $4$
Conductor $12672$
Sign $-1$
Analytic cond. $0.807977$
Root an. cond. $0.948090$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 5·11-s − 2·12-s + 16-s − 18-s − 8·19-s + 5·22-s + 2·24-s − 4·25-s + 4·27-s − 32-s + 10·33-s + 36-s + 8·38-s − 2·43-s − 5·44-s − 2·48-s − 10·49-s + 4·50-s − 4·54-s + 16·57-s − 12·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.577·12-s + 1/4·16-s − 0.235·18-s − 1.83·19-s + 1.06·22-s + 0.408·24-s − 4/5·25-s + 0.769·27-s − 0.176·32-s + 1.74·33-s + 1/6·36-s + 1.29·38-s − 0.304·43-s − 0.753·44-s − 0.288·48-s − 1.42·49-s + 0.565·50-s − 0.544·54-s + 2.11·57-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12672\)    =    \(2^{7} \cdot 3^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(0.807977\)
Root analytic conductor: \(0.948090\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 12672,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 6 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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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

−10.76896165377740111889400970990, −10.72456811555857477059429494106, −10.04670978474008138081387156700, −9.531491561623668367321658818355, −8.627589406228296016139351811637, −8.253461420269860182590760700217, −7.64779178759689521001280835033, −6.92652556351001648987570022036, −6.22621400337090045836497424117, −5.83824467931222926156658261948, −5.03754927864633170100258911165, −4.40193888780434964168241420219, −3.10221168085412076628640338944, −2.04027916243229308840225758835, 0, 2.04027916243229308840225758835, 3.10221168085412076628640338944, 4.40193888780434964168241420219, 5.03754927864633170100258911165, 5.83824467931222926156658261948, 6.22621400337090045836497424117, 6.92652556351001648987570022036, 7.64779178759689521001280835033, 8.253461420269860182590760700217, 8.627589406228296016139351811637, 9.531491561623668367321658818355, 10.04670978474008138081387156700, 10.72456811555857477059429494106, 10.76896165377740111889400970990

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