Properties

Degree $4$
Conductor $327184$
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  + 2·3-s − 3·9-s − 4·13-s + 12·17-s − 6·23-s − 25-s − 14·27-s − 8·39-s − 20·43-s − 10·49-s + 24·51-s − 12·53-s − 8·61-s − 12·69-s − 2·75-s + 4·79-s − 4·81-s + 36·101-s + 16·103-s + 12·107-s − 30·113-s + 12·117-s + 121-s + 127-s − 40·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 1.15·3-s − 9-s − 1.10·13-s + 2.91·17-s − 1.25·23-s − 1/5·25-s − 2.69·27-s − 1.28·39-s − 3.04·43-s − 1.42·49-s + 3.36·51-s − 1.64·53-s − 1.02·61-s − 1.44·69-s − 0.230·75-s + 0.450·79-s − 4/9·81-s + 3.58·101-s + 1.57·103-s + 1.16·107-s − 2.82·113-s + 1.10·117-s + 1/11·121-s + 0.0887·127-s − 3.52·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(327184\)    =    \(2^{4} \cdot 11^{2} \cdot 13^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{327184} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 327184,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
11$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.314187785972322305192087673902, −8.132665807481574488057539990847, −7.66471435961055676315007320566, −7.60852684015633966980596452192, −6.63716263814174422745171385492, −6.11181518853748875342915860937, −5.69510294756229368768383854980, −5.11227132195889762323685903220, −4.77814882467244849114054024293, −3.68139626518860523217158413051, −3.27381985666804460223319703873, −3.10915852533965624890419491672, −2.22738900554088144481773298380, −1.58763508602526038578941432974, 0, 1.58763508602526038578941432974, 2.22738900554088144481773298380, 3.10915852533965624890419491672, 3.27381985666804460223319703873, 3.68139626518860523217158413051, 4.77814882467244849114054024293, 5.11227132195889762323685903220, 5.69510294756229368768383854980, 6.11181518853748875342915860937, 6.63716263814174422745171385492, 7.60852684015633966980596452192, 7.66471435961055676315007320566, 8.132665807481574488057539990847, 8.314187785972322305192087673902

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