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

\(L(\frac{1}{2})\) \(\approx\) \(1.310510870\)
\(L(\frac12)\) \(\approx\) \(1.310510870\)
\(L(1)\) 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

−8.924840331713877732469922041288, −8.526233545765959870534761029644, −8.237971496252668051365799766698, −8.012384872896461317751539076957, −7.32857858289337515784577252081, −7.09624950196721262391253256446, −6.89490832464134106515289714728, −6.22559801106408865588512433282, −5.74601958566405836817432535151, −5.46057489774343533994656622524, −5.32900669016639210065358322191, −4.65177911743131635148143819386, −4.34747779460662295910268534792, −3.83710651073430088809657606669, −3.63399016935119989243603171591, −3.05542926507876727353684843455, −2.48480195480771585385219921642, −2.10842343207064511430585286330, −1.03148092965203080579482246530, −0.898819031395489928031866056968, 0.898819031395489928031866056968, 1.03148092965203080579482246530, 2.10842343207064511430585286330, 2.48480195480771585385219921642, 3.05542926507876727353684843455, 3.63399016935119989243603171591, 3.83710651073430088809657606669, 4.34747779460662295910268534792, 4.65177911743131635148143819386, 5.32900669016639210065358322191, 5.46057489774343533994656622524, 5.74601958566405836817432535151, 6.22559801106408865588512433282, 6.89490832464134106515289714728, 7.09624950196721262391253256446, 7.32857858289337515784577252081, 8.012384872896461317751539076957, 8.237971496252668051365799766698, 8.526233545765959870534761029644, 8.924840331713877732469922041288

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