Properties

Label 4-1216800-1.1-c1e2-0-25
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 − 8-s + 9-s + 16-s − 5·17-s − 18-s + 25-s + 4·29-s − 32-s + 5·34-s + 36-s − 37-s − 16·41-s − 8·49-s − 50-s + 7·53-s − 4·58-s + 14·61-s + 64-s − 5·68-s − 72-s + 5·73-s + 74-s + 81-s + 16·82-s − 2·89-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 1/5·25-s + 0.742·29-s − 0.176·32-s + 0.857·34-s + 1/6·36-s − 0.164·37-s − 2.49·41-s − 8/7·49-s − 0.141·50-s + 0.961·53-s − 0.525·58-s + 1.79·61-s + 1/8·64-s − 0.606·68-s − 0.117·72-s + 0.585·73-s + 0.116·74-s + 1/9·81-s + 1.76·82-s − 0.211·89-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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_2$ \( 1 + p T^{2} \)
good7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−7.81569857650125098439065885692, −7.41967893188920318352026767985, −6.82913000061957012372257152727, −6.59188427270805086448045504340, −6.38823544644026819052666029640, −5.49896144712689598127682265603, −5.20485142635874799828885762524, −4.66056503941909570449604283807, −4.13169647307162551588838959031, −3.55283617539668639272416985298, −3.02073178678657133571294320723, −2.30774136001233637346958815441, −1.85353470478522489447013864991, −1.04629605945313951688018210412, 0, 1.04629605945313951688018210412, 1.85353470478522489447013864991, 2.30774136001233637346958815441, 3.02073178678657133571294320723, 3.55283617539668639272416985298, 4.13169647307162551588838959031, 4.66056503941909570449604283807, 5.20485142635874799828885762524, 5.49896144712689598127682265603, 6.38823544644026819052666029640, 6.59188427270805086448045504340, 6.82913000061957012372257152727, 7.41967893188920318352026767985, 7.81569857650125098439065885692

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