Properties

Label 4-1216800-1.1-c1e2-0-30
Degree $4$
Conductor $1216800$
Sign $-1$
Analytic cond. $77.5842$
Root an. cond. $2.96785$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·5-s − 8-s + 9-s − 2·10-s + 16-s − 10·17-s − 18-s + 2·20-s + 3·25-s − 2·29-s − 32-s + 10·34-s + 36-s − 12·37-s − 2·40-s + 12·41-s + 2·45-s + 2·49-s − 3·50-s + 4·53-s + 2·58-s + 64-s − 10·68-s − 72-s − 6·73-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1/4·16-s − 2.42·17-s − 0.235·18-s + 0.447·20-s + 3/5·25-s − 0.371·29-s − 0.176·32-s + 1.71·34-s + 1/6·36-s − 1.97·37-s − 0.316·40-s + 1.87·41-s + 0.298·45-s + 2/7·49-s − 0.424·50-s + 0.549·53-s + 0.262·58-s + 1/8·64-s − 1.21·68-s − 0.117·72-s − 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1216800\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(77.5842\)
Root analytic conductor: \(2.96785\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1216800,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−7.72399680577781381068190941188, −7.29567887011859489397561277183, −7.00842308966653019977347141560, −6.55866549638180711725462165599, −6.06221011586077769044205992754, −5.84883861903278300730132626923, −5.03941182584389079435190834405, −4.77967253103443767951626263272, −4.11455840147798965310194787051, −3.64250590211989440321490015875, −2.82371961588642408668250267269, −2.23299012576811920414939385951, −1.97420693452990182879932696732, −1.12490882824850953950834997471, 0, 1.12490882824850953950834997471, 1.97420693452990182879932696732, 2.23299012576811920414939385951, 2.82371961588642408668250267269, 3.64250590211989440321490015875, 4.11455840147798965310194787051, 4.77967253103443767951626263272, 5.03941182584389079435190834405, 5.84883861903278300730132626923, 6.06221011586077769044205992754, 6.55866549638180711725462165599, 7.00842308966653019977347141560, 7.29567887011859489397561277183, 7.72399680577781381068190941188

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