Properties

Label 4-623808-1.1-c1e2-0-72
Degree $4$
Conductor $623808$
Sign $-1$
Analytic cond. $39.7745$
Root an. cond. $2.51131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 4·19-s − 6·25-s + 27-s − 12·29-s + 12·41-s + 8·43-s − 14·49-s + 4·53-s − 4·57-s − 8·59-s − 4·61-s − 16·71-s + 20·73-s − 6·75-s + 81-s − 12·87-s + 12·89-s + 24·107-s − 36·113-s − 6·121-s + 12·123-s + 127-s + 8·129-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.917·19-s − 6/5·25-s + 0.192·27-s − 2.22·29-s + 1.87·41-s + 1.21·43-s − 2·49-s + 0.549·53-s − 0.529·57-s − 1.04·59-s − 0.512·61-s − 1.89·71-s + 2.34·73-s − 0.692·75-s + 1/9·81-s − 1.28·87-s + 1.27·89-s + 2.32·107-s − 3.38·113-s − 0.545·121-s + 1.08·123-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(623808\)    =    \(2^{6} \cdot 3^{3} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(39.7745\)
Root analytic conductor: \(2.51131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{623808} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 - T \)
19$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−7.979156449589182873646432658217, −7.62403207119791885504595168937, −7.61950951666595077126724317917, −6.85891827919805546068039331531, −6.14969218484336320739052245813, −6.09791694801381232421198374318, −5.40997773154525098146238432632, −4.83326855247570334999188527992, −4.23309352016035067675133294948, −3.83764998170958613523904255647, −3.37632188765103348554746581671, −2.51659494204614894722398853037, −2.12120802782336302555457408398, −1.36402928363394079803758664889, 0, 1.36402928363394079803758664889, 2.12120802782336302555457408398, 2.51659494204614894722398853037, 3.37632188765103348554746581671, 3.83764998170958613523904255647, 4.23309352016035067675133294948, 4.83326855247570334999188527992, 5.40997773154525098146238432632, 6.09791694801381232421198374318, 6.14969218484336320739052245813, 6.85891827919805546068039331531, 7.61950951666595077126724317917, 7.62403207119791885504595168937, 7.979156449589182873646432658217

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