Properties

Degree $4$
Conductor $12960000$
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  − 2·13-s − 4·37-s − 49-s − 2·61-s + 4·73-s − 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  − 2·13-s − 4·37-s − 49-s − 2·61-s + 4·73-s − 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Motivic weight: \(0\)
Character: induced by $\chi_{3600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12960000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8117459673\)
\(L(\frac12)\) \(\approx\) \(0.8117459673\)
\(L(1)\) 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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2$ \( ( 1 + T + T^{2} )^{2} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$ \( ( 1 + T )^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T + T^{2} )^{2} \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_1$ \( ( 1 - T )^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + T^{2} )^{2} \)
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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

−8.980017735857843812538456317398, −8.495906941142676337876323612065, −8.211667345403131720149641876132, −7.70009629641350050643809782021, −7.51034146867927298879897266666, −7.02991515684197995930810338388, −6.64105846156008902713675095987, −6.62151177421089918367420348919, −5.83527989471391811188545608707, −5.40928291195063443796998300218, −5.14818961518955227816194293011, −4.79065477133777642667361615446, −4.48193437396371786389006378033, −3.80687696692475507210282410059, −3.40572694725035332119634689387, −3.04733404124779388086795473369, −2.49080917123977941079809505530, −1.80936154711806098453555856754, −1.76418639871973891381928831482, −0.49691925903654501326642723863, 0.49691925903654501326642723863, 1.76418639871973891381928831482, 1.80936154711806098453555856754, 2.49080917123977941079809505530, 3.04733404124779388086795473369, 3.40572694725035332119634689387, 3.80687696692475507210282410059, 4.48193437396371786389006378033, 4.79065477133777642667361615446, 5.14818961518955227816194293011, 5.40928291195063443796998300218, 5.83527989471391811188545608707, 6.62151177421089918367420348919, 6.64105846156008902713675095987, 7.02991515684197995930810338388, 7.51034146867927298879897266666, 7.70009629641350050643809782021, 8.211667345403131720149641876132, 8.495906941142676337876323612065, 8.980017735857843812538456317398

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