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Properties

Label 4-1216e2-1.1-c1e2-0-23
Degree $4$
Conductor $1478656$
Sign $-1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·9-s + 2·13-s + 6·17-s − 10·25-s − 6·29-s + 20·37-s − 24·41-s − 13·49-s + 18·53-s − 20·61-s + 2·73-s − 8·89-s − 12·97-s − 20·101-s − 18·109-s + 4·113-s + 6·117-s − 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 18·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 9-s + 0.554·13-s + 1.45·17-s − 2·25-s − 1.11·29-s + 3.28·37-s − 3.74·41-s − 1.85·49-s + 2.47·53-s − 2.56·61-s + 0.234·73-s − 0.847·89-s − 1.21·97-s − 1.99·101-s − 1.72·109-s + 0.376·113-s + 0.554·117-s − 1.63·121-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 + 1.45·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.3.a_ad
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.a_n
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.41.y_is
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.59.a_dp
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.a_dh
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.73.ac_fr
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.89.i_hm
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.74236222044461134007095453218, −7.47536575646542499937872009352, −6.61303503651954034437075836396, −6.60845811420238685670203067170, −5.91824238738469752095279165720, −5.46776906443032883183491997236, −5.22289133265133441977616849928, −4.39011552338323920878309071486, −4.08931083553499496867238827018, −3.63803508541238097399374767207, −3.11827862407422559284554596552, −2.43573806798025905726670590752, −1.52291178817025335850558392010, −1.37942708011457790071691730372, 0, 1.37942708011457790071691730372, 1.52291178817025335850558392010, 2.43573806798025905726670590752, 3.11827862407422559284554596552, 3.63803508541238097399374767207, 4.08931083553499496867238827018, 4.39011552338323920878309071486, 5.22289133265133441977616849928, 5.46776906443032883183491997236, 5.91824238738469752095279165720, 6.60845811420238685670203067170, 6.61303503651954034437075836396, 7.47536575646542499937872009352, 7.74236222044461134007095453218

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