Properties

Degree $4$
Conductor $790272$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 9-s + 8·11-s − 16·23-s − 6·25-s − 4·29-s − 20·37-s + 8·43-s + 49-s + 12·53-s + 63-s − 8·67-s − 16·71-s + 8·77-s + 81-s + 8·99-s − 24·107-s − 4·109-s − 28·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 0.377·7-s + 1/3·9-s + 2.41·11-s − 3.33·23-s − 6/5·25-s − 0.742·29-s − 3.28·37-s + 1.21·43-s + 1/7·49-s + 1.64·53-s + 0.125·63-s − 0.977·67-s − 1.89·71-s + 0.911·77-s + 1/9·81-s + 0.804·99-s − 2.32·107-s − 0.383·109-s − 2.63·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(790272\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{790272} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 790272,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 - T \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.963562405403294064353399542187, −7.67454257840429899018734503053, −6.92538722224681151756066962499, −6.89499765960031010222506198709, −6.18874160468740625251832036222, −5.69761771888975023011122238972, −5.60435712531698215962545180754, −4.60067949543777570406002885677, −4.04752481141371955267217260485, −3.91638973375108536412463827494, −3.54052974946578322805924483948, −2.39883079209175985717649788244, −1.69840313204406173914058397816, −1.50800619952171259085588630937, 0, 1.50800619952171259085588630937, 1.69840313204406173914058397816, 2.39883079209175985717649788244, 3.54052974946578322805924483948, 3.91638973375108536412463827494, 4.04752481141371955267217260485, 4.60067949543777570406002885677, 5.60435712531698215962545180754, 5.69761771888975023011122238972, 6.18874160468740625251832036222, 6.89499765960031010222506198709, 6.92538722224681151756066962499, 7.67454257840429899018734503053, 7.963562405403294064353399542187

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