Properties

Label 4-338688-1.1-c1e2-0-56
Degree $4$
Conductor $338688$
Sign $-1$
Analytic cond. $21.5950$
Root an. cond. $2.15570$
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  − 3-s + 2·7-s + 9-s − 4·13-s − 8·19-s − 2·21-s − 6·25-s − 27-s + 12·37-s + 4·39-s + 8·43-s + 3·49-s + 8·57-s − 4·61-s + 2·63-s − 8·67-s − 12·73-s + 6·75-s + 32·79-s + 81-s − 8·91-s + 36·97-s − 16·103-s − 36·109-s − 12·111-s − 4·117-s − 6·121-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.10·13-s − 1.83·19-s − 0.436·21-s − 6/5·25-s − 0.192·27-s + 1.97·37-s + 0.640·39-s + 1.21·43-s + 3/7·49-s + 1.05·57-s − 0.512·61-s + 0.251·63-s − 0.977·67-s − 1.40·73-s + 0.692·75-s + 3.60·79-s + 1/9·81-s − 0.838·91-s + 3.65·97-s − 1.57·103-s − 3.44·109-s − 1.13·111-s − 0.369·117-s − 0.545·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(338688\)    =    \(2^{8} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(21.5950\)
Root analytic conductor: \(2.15570\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 + T \)
7$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−8.351153010775658504468507691128, −8.068098981215555715874277637796, −7.45452630662509700459332399892, −7.38325958034671995701748313853, −6.43539728173776997986575871832, −6.19643457342138342025418009798, −5.76945209288606009941542010236, −4.94701466488767178340177905762, −4.75834670654208540745088589071, −4.13981751462080886088964913424, −3.71279869000131289976682575519, −2.39234840548789475685588947322, −2.38203049551357769485459868076, −1.27003349896142462152543818564, 0, 1.27003349896142462152543818564, 2.38203049551357769485459868076, 2.39234840548789475685588947322, 3.71279869000131289976682575519, 4.13981751462080886088964913424, 4.75834670654208540745088589071, 4.94701466488767178340177905762, 5.76945209288606009941542010236, 6.19643457342138342025418009798, 6.43539728173776997986575871832, 7.38325958034671995701748313853, 7.45452630662509700459332399892, 8.068098981215555715874277637796, 8.351153010775658504468507691128

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