↑

Properties

Label 4-208e2-1.1-c1e2-0-13
Degree $4$
Conductor $43264$
Sign $-1$
Analytic cond. $2.75855$
Root an. cond. $1.28875$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 9-s + 6·17-s + 3·25-s − 8·29-s − 12·37-s − 8·41-s + 4·45-s + 5·49-s − 4·53-s − 16·61-s − 16·73-s − 8·81-s − 24·85-s − 8·89-s + 4·97-s + 8·101-s + 4·109-s + 20·113-s + 10·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + ⋯
L(s)  = 1  − 1.78·5-s − 1/3·9-s + 1.45·17-s + 3/5·25-s − 1.48·29-s − 1.97·37-s − 1.24·41-s + 0.596·45-s + 5/7·49-s − 0.549·53-s − 2.04·61-s − 1.87·73-s − 8/9·81-s − 2.60·85-s − 0.847·89-s + 0.406·97-s + 0.796·101-s + 0.383·109-s + 1.88·113-s + 0.909·121-s + 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(43264\)    =    \(2^{8} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(2.75855\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 43264,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
5$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.e_n
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.17.ag_bb
19$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.19.a_as
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.i_cs
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.31.a_bi
37$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.m_dx
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.i_bi
43$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.43.a_ax
47$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \) 2.47.a_cp
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.e_ec
59$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.59.a_abi
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.67.a_di
71$C_2^2$ \( 1 + 123 T^{2} + p^{2} T^{4} \) 2.71.a_et
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.q_hy
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 - 134 T^{2} + p^{2} T^{4} \) 2.83.a_afe
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.i_dq
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ae_ha
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

−10.05388583804179182454648122016, −9.418255782270657682104939128553, −8.735572786889196785225324425288, −8.388820351083467531115469238700, −7.80490515923189006865582892845, −7.32779465242594216094746402162, −7.14065374730245462049984933610, −6.04618129714894847182353682080, −5.60958235390271557851881078821, −4.86071352318496454453301688251, −4.16931961393237786703767039878, −3.45548671744298782195200490697, −3.21426389289253930949098701996, −1.71639683162303224663633428360, 0, 1.71639683162303224663633428360, 3.21426389289253930949098701996, 3.45548671744298782195200490697, 4.16931961393237786703767039878, 4.86071352318496454453301688251, 5.60958235390271557851881078821, 6.04618129714894847182353682080, 7.14065374730245462049984933610, 7.32779465242594216094746402162, 7.80490515923189006865582892845, 8.388820351083467531115469238700, 8.735572786889196785225324425288, 9.418255782270657682104939128553, 10.05388583804179182454648122016

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