Properties

Label 4-448e2-1.1-c1e2-0-44
Degree $4$
Conductor $200704$
Sign $1$
Analytic cond. $12.7970$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 6·9-s − 4·13-s − 12·17-s + 2·25-s − 12·29-s + 4·37-s + 4·41-s + 24·45-s + 49-s − 12·53-s + 12·61-s + 16·65-s + 20·73-s + 27·81-s + 48·85-s − 12·89-s − 12·97-s − 4·101-s + 20·109-s + 4·113-s + 24·117-s − 6·121-s + 28·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.78·5-s − 2·9-s − 1.10·13-s − 2.91·17-s + 2/5·25-s − 2.22·29-s + 0.657·37-s + 0.624·41-s + 3.57·45-s + 1/7·49-s − 1.64·53-s + 1.53·61-s + 1.98·65-s + 2.34·73-s + 3·81-s + 5.20·85-s − 1.27·89-s − 1.21·97-s − 0.398·101-s + 1.91·109-s + 0.376·113-s + 2.21·117-s − 0.545·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(200704\)    =    \(2^{12} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(12.7970\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{200704} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 200704,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.749304120934004278681759852831, −8.056242270008326333596348707107, −7.79292083571821938290976650763, −7.26382021666693896035142333427, −6.78746217059840482908609994194, −6.15054397459112152814277916997, −5.69394971603405508863657314834, −4.92225264716551837532790929786, −4.58367495132727840614913978641, −3.77515777589665000060106383714, −3.60173413237973468362473708205, −2.40145156924384381758797773878, −2.35911554344998475560129433401, 0, 0, 2.35911554344998475560129433401, 2.40145156924384381758797773878, 3.60173413237973468362473708205, 3.77515777589665000060106383714, 4.58367495132727840614913978641, 4.92225264716551837532790929786, 5.69394971603405508863657314834, 6.15054397459112152814277916997, 6.78746217059840482908609994194, 7.26382021666693896035142333427, 7.79292083571821938290976650763, 8.056242270008326333596348707107, 8.749304120934004278681759852831

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