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  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·17-s + 2·19-s − 2·23-s − 6·32-s − 4·34-s − 4·38-s + 4·46-s + 2·47-s + 2·61-s + 7·64-s + 6·68-s + 6·76-s + 4·83-s − 6·92-s − 4·94-s + 4·107-s − 2·109-s − 2·113-s − 4·122-s + 127-s − 8·128-s + 131-s − 8·136-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 2·17-s + 2·19-s − 2·23-s − 6·32-s − 4·34-s − 4·38-s + 4·46-s + 2·47-s + 2·61-s + 7·64-s + 6·68-s + 6·76-s + 4·83-s − 6·92-s − 4·94-s + 4·107-s − 2·109-s − 2·113-s − 4·122-s + 127-s − 8·128-s + 131-s − 8·136-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

\(L(\frac{1}{2})\) \(\approx\) \(0.7179313801\)
\(L(\frac12)\) \(\approx\) \(0.7179313801\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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$C_1$ \( ( 1 - T )^{4} \)
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

−8.763907658950581860575461707022, −8.747506268940613382712887968863, −7.949276505332593696836538162918, −7.896773680884115952924095968678, −7.54005616866590754321907977609, −7.49948590138733128515487477591, −6.75788248891037997492784621781, −6.55877104087657494037607875114, −5.96427359175341310717797828612, −5.73867384980883470554502285022, −5.33240190214868689701780887518, −5.08251563771190697598871274012, −3.90029188327220822394473588323, −3.85240628417037585804803315095, −3.12711639803258165308250889241, −2.99415304506268720952436637574, −2.05766774731570020841204835253, −2.05762052439398049788431617748, −1.00039767115762042892346774083, −0.901534009153222065044194060008, 0.901534009153222065044194060008, 1.00039767115762042892346774083, 2.05762052439398049788431617748, 2.05766774731570020841204835253, 2.99415304506268720952436637574, 3.12711639803258165308250889241, 3.85240628417037585804803315095, 3.90029188327220822394473588323, 5.08251563771190697598871274012, 5.33240190214868689701780887518, 5.73867384980883470554502285022, 5.96427359175341310717797828612, 6.55877104087657494037607875114, 6.75788248891037997492784621781, 7.49948590138733128515487477591, 7.54005616866590754321907977609, 7.896773680884115952924095968678, 7.949276505332593696836538162918, 8.747506268940613382712887968863, 8.763907658950581860575461707022

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