↑

Properties

Label 4-2288e2-1.1-c1e2-0-10
Degree $4$
Conductor $5234944$
Sign $1$
Analytic cond. $333.784$
Root an. cond. $4.27431$
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  − 2·3-s − 6·5-s − 3·9-s − 6·11-s + 12·15-s + 17·25-s + 14·27-s + 8·31-s + 12·33-s − 14·37-s + 18·45-s − 6·47-s − 13·49-s + 36·55-s + 12·59-s − 28·67-s + 6·71-s − 34·75-s − 4·81-s − 12·89-s − 16·93-s − 20·97-s + 18·99-s + 8·103-s + 28·111-s − 12·113-s + 25·121-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.68·5-s − 9-s − 1.80·11-s + 3.09·15-s + 17/5·25-s + 2.69·27-s + 1.43·31-s + 2.08·33-s − 2.30·37-s + 2.68·45-s − 0.875·47-s − 1.85·49-s + 4.85·55-s + 1.56·59-s − 3.42·67-s + 0.712·71-s − 3.92·75-s − 4/9·81-s − 1.27·89-s − 1.65·93-s − 2.03·97-s + 1.80·99-s + 0.788·103-s + 2.65·111-s − 1.12·113-s + 2.27·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5234944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5234944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5234944\)    =    \(2^{8} \cdot 11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(333.784\)
Root analytic conductor: \(4.27431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5234944,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.a_z
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.37.o_et
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.a_dh
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.67.bc_ms
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.71.ag_fv
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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

−6.80992153386544904186537940607, −6.66224713551990484880669593056, −6.03737998329952530382717734620, −5.41734923280427435562979846122, −5.35854408369170434929735266448, −4.83697201475855068833075570215, −4.44297288531505612446396437446, −4.08008544759795861745759386080, −3.31636164373489769760153942339, −3.03319543886757601365262705803, −2.86140175572535291493434732878, −1.83467892554345205053408946219, −0.71484850063807726614584220761, 0, 0, 0.71484850063807726614584220761, 1.83467892554345205053408946219, 2.86140175572535291493434732878, 3.03319543886757601365262705803, 3.31636164373489769760153942339, 4.08008544759795861745759386080, 4.44297288531505612446396437446, 4.83697201475855068833075570215, 5.35854408369170434929735266448, 5.41734923280427435562979846122, 6.03737998329952530382717734620, 6.66224713551990484880669593056, 6.80992153386544904186537940607

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