Properties

Degree $4$
Conductor $42336$
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  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 8·11-s + 12-s + 12·13-s + 16-s − 18-s − 8·22-s − 16·23-s − 24-s − 6·25-s − 12·26-s + 27-s − 32-s + 8·33-s + 36-s − 20·37-s + 12·39-s + 8·44-s + 16·46-s + 48-s + 49-s + 6·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 2.41·11-s + 0.288·12-s + 3.32·13-s + 1/4·16-s − 0.235·18-s − 1.70·22-s − 3.33·23-s − 0.204·24-s − 6/5·25-s − 2.35·26-s + 0.192·27-s − 0.176·32-s + 1.39·33-s + 1/6·36-s − 3.28·37-s + 1.92·39-s + 1.20·44-s + 2.35·46-s + 0.144·48-s + 1/7·49-s + 0.848·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(42336\)    =    \(2^{5} \cdot 3^{3} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{42336} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 42336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.435805038\)
\(L(\frac12)\) \(\approx\) \(1.435805038\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.13805865622749455803240285872, −9.567727627901464767318320243097, −9.164662289745378790951914691450, −8.579197535720292899748335278003, −8.337652051623380123011624806396, −7.963562405403294064353399542187, −6.89499765960031010222506198709, −6.58813712157544804919343144385, −6.01899647743800981343957937964, −5.69761771888975023011122238972, −4.04752481141371955267217260485, −3.83863171110134241641513411613, −3.54052974946578322805924483948, −1.79365373540820646322667624458, −1.50800619952171259085588630937, 1.50800619952171259085588630937, 1.79365373540820646322667624458, 3.54052974946578322805924483948, 3.83863171110134241641513411613, 4.04752481141371955267217260485, 5.69761771888975023011122238972, 6.01899647743800981343957937964, 6.58813712157544804919343144385, 6.89499765960031010222506198709, 7.963562405403294064353399542187, 8.337652051623380123011624806396, 8.579197535720292899748335278003, 9.164662289745378790951914691450, 9.567727627901464767318320243097, 10.13805865622749455803240285872

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