Properties

Degree $4$
Conductor $12446784$
Sign $1$
Motivic weight $0$
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 − 6-s + 8-s + 11-s − 16-s + 2·17-s + 2·19-s − 22-s + 24-s − 25-s − 27-s + 33-s − 2·34-s − 2·38-s − 41-s + 43-s − 48-s + 50-s + 2·51-s + 54-s + 2·57-s − 59-s + 64-s − 66-s + 67-s + 2·73-s + ⋯
L(s)  = 1  − 2-s + 3-s − 6-s + 8-s + 11-s − 16-s + 2·17-s + 2·19-s − 22-s + 24-s − 25-s − 27-s + 33-s − 2·34-s − 2·38-s − 41-s + 43-s − 48-s + 50-s + 2·51-s + 54-s + 2·57-s − 59-s + 64-s − 66-s + 67-s + 2·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12446784\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{3528} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12446784,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.319634231\)
\(L(\frac12)\) \(\approx\) \(1.319634231\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
17$C_2$ \( ( 1 - T + T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )^{2} \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 - T + T^{2} )^{2} \)
89$C_1$ \( ( 1 + T )^{4} \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
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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

−9.142792477057806123827918849226, −8.486533330801220667603427185049, −8.114783648399841316673735849006, −7.896955186343717929773410806403, −7.81293167792934109801927586975, −7.26958983436783143438935801586, −6.82424923555793465258863066796, −6.65082042375633154700854803211, −5.74777021942009769818086655632, −5.56120935146871348339114457007, −5.36029001331115272600287279806, −4.72125741956824816919274408551, −4.04260115477821288844358963984, −3.91513948962730953183343170928, −3.26256151468185372240320607177, −3.16202888327544439705383499544, −2.47221110691323698640157847026, −1.78387239858480799404086600067, −1.37474811863420541552851100091, −0.853757497419021379698111168003, 0.853757497419021379698111168003, 1.37474811863420541552851100091, 1.78387239858480799404086600067, 2.47221110691323698640157847026, 3.16202888327544439705383499544, 3.26256151468185372240320607177, 3.91513948962730953183343170928, 4.04260115477821288844358963984, 4.72125741956824816919274408551, 5.36029001331115272600287279806, 5.56120935146871348339114457007, 5.74777021942009769818086655632, 6.65082042375633154700854803211, 6.82424923555793465258863066796, 7.26958983436783143438935801586, 7.81293167792934109801927586975, 7.896955186343717929773410806403, 8.114783648399841316673735849006, 8.486533330801220667603427185049, 9.142792477057806123827918849226

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