Properties

Degree $4$
Conductor $12446784$
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  + 2·5-s + 12·13-s − 2·17-s − 4·19-s − 4·23-s + 5·25-s + 20·29-s + 8·31-s − 6·37-s + 4·41-s − 8·43-s + 8·47-s − 10·53-s + 12·59-s + 2·61-s + 24·65-s − 12·67-s + 24·71-s + 14·73-s + 8·79-s − 24·83-s − 4·85-s − 2·89-s − 8·95-s + 20·97-s − 6·101-s + 2·109-s + ⋯
L(s)  = 1  + 0.894·5-s + 3.32·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 25-s + 3.71·29-s + 1.43·31-s − 0.986·37-s + 0.624·41-s − 1.21·43-s + 1.16·47-s − 1.37·53-s + 1.56·59-s + 0.256·61-s + 2.97·65-s − 1.46·67-s + 2.84·71-s + 1.63·73-s + 0.900·79-s − 2.63·83-s − 0.433·85-s − 0.211·89-s − 0.820·95-s + 2.03·97-s − 0.597·101-s + 0.191·109-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(5.127702942\)
\(L(\frac12)\) \(\approx\) \(5.127702942\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−8.502822641816571771634306320565, −8.424920741814522036659944694045, −8.218986338123309159730323495793, −8.022155646319705380616939473410, −6.87736678257361120134973727372, −6.83372796223109581336696846951, −6.36788753780212138931747754294, −6.34310986327447485074712883525, −5.88001750581901972782425508733, −5.50272507745461089850190269729, −4.83657720312791131093659864617, −4.64780565590878982475590182341, −3.94983628381681533987193806165, −3.90004202929853511830446563241, −3.07618351361672951769536253124, −2.91546801257104559277777596806, −2.18618027198153737951585901659, −1.75605530083727811500754861500, −1.00277133427536567828942054591, −0.872398733712217342574418147342, 0.872398733712217342574418147342, 1.00277133427536567828942054591, 1.75605530083727811500754861500, 2.18618027198153737951585901659, 2.91546801257104559277777596806, 3.07618351361672951769536253124, 3.90004202929853511830446563241, 3.94983628381681533987193806165, 4.64780565590878982475590182341, 4.83657720312791131093659864617, 5.50272507745461089850190269729, 5.88001750581901972782425508733, 6.34310986327447485074712883525, 6.36788753780212138931747754294, 6.83372796223109581336696846951, 6.87736678257361120134973727372, 8.022155646319705380616939473410, 8.218986338123309159730323495793, 8.424920741814522036659944694045, 8.502822641816571771634306320565

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