↑

Properties

Label 4-338688-1.1-c1e2-0-28
Degree $4$
Conductor $338688$
Sign $1$
Analytic cond. $21.5950$
Root an. cond. $2.15570$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s + 9-s − 4·19-s + 2·21-s + 6·25-s + 27-s − 6·29-s + 12·31-s + 4·37-s + 4·47-s − 3·49-s + 2·53-s − 4·57-s + 12·59-s + 2·63-s + 6·75-s + 81-s − 16·83-s − 6·87-s + 12·93-s − 4·103-s + 4·111-s + 2·113-s + 2·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.917·19-s + 0.436·21-s + 6/5·25-s + 0.192·27-s − 1.11·29-s + 2.15·31-s + 0.657·37-s + 0.583·47-s − 3/7·49-s + 0.274·53-s − 0.529·57-s + 1.56·59-s + 0.251·63-s + 0.692·75-s + 1/9·81-s − 1.75·83-s − 0.643·87-s + 1.24·93-s − 0.394·103-s + 0.379·111-s + 0.188·113-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(338688\)    =    \(2^{8} \cdot 3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(21.5950\)
Root analytic conductor: \(2.15570\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.544970041\)
\(L(\frac12)\) \(\approx\) \(2.544970041\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( 1 - T \)
7$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.23.a_w
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.g_co
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.am_dq
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ae_ck
41$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.41.a_by
43$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.43.a_ack
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.ae_ck
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.53.ac_ec
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.am_eo
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.67.a_di
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.71.a_be
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.a_de
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.79.a_da
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.q_ig
89$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.89.a_acs
97$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.97.a_ek
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

−8.555002189705458654869928996261, −8.316618033431768854591499790598, −8.063332402052162939100193945194, −7.31987402829276542831799625511, −7.00616485699272467774806186278, −6.48423237759858608774282326564, −5.91031591508502853233310818870, −5.38615572186922390524852344216, −4.68951589441240668548334387524, −4.41157572424766078704073590205, −3.83312899138907642541696558962, −3.03801687984380639633195562028, −2.51962272467889600085657593418, −1.83887034870592950817785607013, −0.931041842329516825417157384683, 0.931041842329516825417157384683, 1.83887034870592950817785607013, 2.51962272467889600085657593418, 3.03801687984380639633195562028, 3.83312899138907642541696558962, 4.41157572424766078704073590205, 4.68951589441240668548334387524, 5.38615572186922390524852344216, 5.91031591508502853233310818870, 6.48423237759858608774282326564, 7.00616485699272467774806186278, 7.31987402829276542831799625511, 8.063332402052162939100193945194, 8.316618033431768854591499790598, 8.555002189705458654869928996261

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