↑

Properties

Label 4-2376e2-1.1-c1e2-0-12
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
Motivic weight $1$
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  + 8·5-s + 4·11-s + 8·23-s + 38·25-s − 8·31-s − 18·37-s − 24·47-s − 5·49-s − 16·53-s + 32·55-s + 8·59-s + 22·67-s + 16·71-s + 24·89-s + 10·97-s + 2·103-s + 24·113-s + 64·115-s + 5·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·155-s + ⋯
L(s)  = 1  + 3.57·5-s + 1.20·11-s + 1.66·23-s + 38/5·25-s − 1.43·31-s − 2.95·37-s − 3.50·47-s − 5/7·49-s − 2.19·53-s + 4.31·55-s + 1.04·59-s + 2.68·67-s + 1.89·71-s + 2.54·89-s + 1.01·97-s + 0.197·103-s + 2.25·113-s + 5.96·115-s + 5/11·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.708444023\)
\(L(\frac12)\) \(\approx\) \(6.708444023\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.5.ai_ba
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.a_bl
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.37.s_fz
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.53.q_go
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.61.a_dt
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.73.a_fp
79$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.79.a_fd
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.89.ay_mk
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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

−6.85949407456534069359160628539, −6.65045650092432229207532369718, −6.44248673632066345518308551587, −6.29322372363163853531675191252, −5.50877258085363456321185078726, −5.18337970194542600248971887273, −5.10755730919050596352378085939, −4.76803609235568144651732056357, −3.64054923239624319126091646659, −3.36478379045069074059258347613, −2.92588943026385905228759535472, −2.06553802195663976566119194064, −1.81832624891237728767312244270, −1.65095305228095896172506485539, −0.885735380022343869140855994455, 0.885735380022343869140855994455, 1.65095305228095896172506485539, 1.81832624891237728767312244270, 2.06553802195663976566119194064, 2.92588943026385905228759535472, 3.36478379045069074059258347613, 3.64054923239624319126091646659, 4.76803609235568144651732056357, 5.10755730919050596352378085939, 5.18337970194542600248971887273, 5.50877258085363456321185078726, 6.29322372363163853531675191252, 6.44248673632066345518308551587, 6.65045650092432229207532369718, 6.85949407456534069359160628539

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