Properties

Label 4-189728-1.1-c1e2-0-13
Degree $4$
Conductor $189728$
Sign $1$
Analytic cond. $12.0972$
Root an. cond. $1.86496$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 4·7-s + 8-s + 2·9-s + 4·14-s + 16-s + 2·18-s + 2·25-s + 4·28-s + 32-s + 2·36-s + 8·37-s − 8·43-s + 9·49-s + 2·50-s + 4·56-s + 8·63-s + 64-s − 12·67-s − 12·71-s + 2·72-s + 8·74-s − 8·79-s − 5·81-s − 8·86-s + 9·98-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 2/3·9-s + 1.06·14-s + 1/4·16-s + 0.471·18-s + 2/5·25-s + 0.755·28-s + 0.176·32-s + 1/3·36-s + 1.31·37-s − 1.21·43-s + 9/7·49-s + 0.282·50-s + 0.534·56-s + 1.00·63-s + 1/8·64-s − 1.46·67-s − 1.42·71-s + 0.235·72-s + 0.929·74-s − 0.900·79-s − 5/9·81-s − 0.862·86-s + 0.909·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(189728\)    =    \(2^{5} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(12.0972\)
Root analytic conductor: \(1.86496\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 189728,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.447989840\)
\(L(\frac12)\) \(\approx\) \(3.447989840\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.906882131258336079135241598201, −8.692390619681186858541601156191, −8.053774037860861526284803484305, −7.60876951261413716282408568292, −7.28318251391559408018335136270, −6.71418582032898676080660671172, −6.06039846026110002903309874803, −5.64467030299328163838244751194, −4.93751385579225310082311837843, −4.58677346592566655594097328486, −4.22673477612549840219210110383, −3.41747970398511736530605125393, −2.69499092665128003067683925742, −1.87032529301807589733451855292, −1.24870221776196531379907341405, 1.24870221776196531379907341405, 1.87032529301807589733451855292, 2.69499092665128003067683925742, 3.41747970398511736530605125393, 4.22673477612549840219210110383, 4.58677346592566655594097328486, 4.93751385579225310082311837843, 5.64467030299328163838244751194, 6.06039846026110002903309874803, 6.71418582032898676080660671172, 7.28318251391559408018335136270, 7.60876951261413716282408568292, 8.053774037860861526284803484305, 8.692390619681186858541601156191, 8.906882131258336079135241598201

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