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  − 4-s + 16-s − 2·17-s − 2·19-s − 2·23-s − 2·47-s − 4·53-s + 2·61-s − 64-s + 2·68-s + 2·76-s + 2·92-s + 2·109-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 4-s + 16-s − 2·17-s − 2·19-s − 2·23-s − 2·47-s − 4·53-s + 2·61-s − 64-s + 2·68-s + 2·76-s + 2·92-s + 2·109-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-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

\(L(\frac{1}{2})\) \(\approx\) \(0.3231541701\)
\(L(\frac12)\) \(\approx\) \(0.3231541701\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
53$C_1$ \( ( 1 + T )^{4} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−8.840329471585470289009577905146, −8.464483206693136692419725031041, −8.307954698974186515567994325747, −7.950447178398915132708878322573, −7.64546647484322424397325663200, −6.88925762661635178074584972853, −6.61661409666853980558975589704, −6.40424146961632750196400319984, −5.83733642954096286456462120135, −5.75072337935010036189406773612, −4.80637539491969613091654765738, −4.69870004726746144078376948285, −4.43682765824532894580321547985, −4.04885870075158876883996776906, −3.43220991952788035647079883045, −3.21988968857554125625413749981, −2.21963929158248161733974633278, −2.09847707073508280553719848146, −1.56324844890515915473163182591, −0.32011811031841551413749580584, 0.32011811031841551413749580584, 1.56324844890515915473163182591, 2.09847707073508280553719848146, 2.21963929158248161733974633278, 3.21988968857554125625413749981, 3.43220991952788035647079883045, 4.04885870075158876883996776906, 4.43682765824532894580321547985, 4.69870004726746144078376948285, 4.80637539491969613091654765738, 5.75072337935010036189406773612, 5.83733642954096286456462120135, 6.40424146961632750196400319984, 6.61661409666853980558975589704, 6.88925762661635178074584972853, 7.64546647484322424397325663200, 7.950447178398915132708878322573, 8.307954698974186515567994325747, 8.464483206693136692419725031041, 8.840329471585470289009577905146

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