↑

Properties

Label 4-583e2-1.1-c0e2-0-0
Degree $4$
Conductor $339889$
Sign $1$
Analytic cond. $0.0846547$
Root an. cond. $0.539402$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 2·11-s − 16-s + 2·25-s + 2·49-s − 2·53-s + 3·81-s − 4·97-s − 4·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 2·176-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2·9-s − 2·11-s − 16-s + 2·25-s + 2·49-s − 2·53-s + 3·81-s − 4·97-s − 4·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 2·176-s + 179-s + 181-s + 191-s + 193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 339889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 339889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(339889\)    =    \(11^{2} \cdot 53^{2}\)
Sign: $1$
Analytic conductor: \(0.0846547\)
Root analytic conductor: \(0.539402\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 339889,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8288205342\)
\(L(\frac12)\) \(\approx\) \(0.8288205342\)
\(L(1)\) 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)$
bad11$C_1$ \( ( 1 + T )^{2} \)
53$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2$ \( ( 1 + T^{2} )^{2} \)
97$C_1$ \( ( 1 + T )^{4} \)
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

−10.95588267360171084045005274293, −10.60994472575109254570439166882, −10.37472168028696448811022258221, −9.941893276077822530522270628546, −9.292167414402185481710314062044, −9.203373432392943991133142709438, −8.309488400626005141313969521472, −8.129175828008130593857409960336, −7.49992263619999484278332216324, −7.00483697264851107501596750842, −6.94730329874278608003273366711, −6.25149995737344638373087432276, −5.47201978361553177109939172354, −5.10185462377440965431208669523, −4.47592467563617266270253980810, −4.35585707442430047402072272131, −3.39768816516866423145709770700, −2.72674998786741126297726161667, −2.18552916758272898251961246630, −1.24145833078272359533649344522, 1.24145833078272359533649344522, 2.18552916758272898251961246630, 2.72674998786741126297726161667, 3.39768816516866423145709770700, 4.35585707442430047402072272131, 4.47592467563617266270253980810, 5.10185462377440965431208669523, 5.47201978361553177109939172354, 6.25149995737344638373087432276, 6.94730329874278608003273366711, 7.00483697264851107501596750842, 7.49992263619999484278332216324, 8.129175828008130593857409960336, 8.309488400626005141313969521472, 9.203373432392943991133142709438, 9.292167414402185481710314062044, 9.941893276077822530522270628546, 10.37472168028696448811022258221, 10.60994472575109254570439166882, 10.95588267360171084045005274293

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