Properties

Degree $4$
Conductor $338688$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s + 9-s + 12·13-s + 8·19-s + 2·21-s − 6·25-s + 27-s − 20·37-s + 12·39-s + 8·43-s + 3·49-s + 8·57-s + 12·61-s + 2·63-s − 8·67-s + 20·73-s − 6·75-s + 81-s + 24·91-s − 28·97-s − 16·103-s − 4·109-s − 20·111-s + 12·117-s − 6·121-s + 127-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 3.32·13-s + 1.83·19-s + 0.436·21-s − 6/5·25-s + 0.192·27-s − 3.28·37-s + 1.92·39-s + 1.21·43-s + 3/7·49-s + 1.05·57-s + 1.53·61-s + 0.251·63-s − 0.977·67-s + 2.34·73-s − 0.692·75-s + 1/9·81-s + 2.51·91-s − 2.84·97-s − 1.57·103-s − 0.383·109-s − 1.89·111-s + 1.10·117-s − 0.545·121-s + 0.0887·127-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: $1$
Motivic weight: \(1\)
Character: $\chi_{338688} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.164303633\)
\(L(\frac12)\) \(\approx\) \(3.164303633\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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

−8.579197535720292899748335278003, −8.427817591055194605299416158333, −7.963562405403294064353399542187, −7.46998199559376075566652373057, −6.89499765960031010222506198709, −6.48878700424638503015501865218, −5.69761771888975023011122238972, −5.58110052520146908580921209772, −4.96240850233932644050318806427, −4.04752481141371955267217260485, −3.57539886636491016875562737772, −3.54052974946578322805924483948, −2.47676327970158091646809943718, −1.50800619952171259085588630937, −1.22168166913217334012046287130, 1.22168166913217334012046287130, 1.50800619952171259085588630937, 2.47676327970158091646809943718, 3.54052974946578322805924483948, 3.57539886636491016875562737772, 4.04752481141371955267217260485, 4.96240850233932644050318806427, 5.58110052520146908580921209772, 5.69761771888975023011122238972, 6.48878700424638503015501865218, 6.89499765960031010222506198709, 7.46998199559376075566652373057, 7.963562405403294064353399542187, 8.427817591055194605299416158333, 8.579197535720292899748335278003

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