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Properties

Label 4-987696-1.1-c1e2-0-5
Degree $4$
Conductor $987696$
Sign $-1$
Analytic cond. $62.9763$
Root an. cond. $2.81704$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·7-s − 3·9-s + 19-s + 3·25-s + 2·43-s + 13·49-s + 14·61-s + 18·63-s − 6·73-s + 9·81-s + 15·121-s + 127-s + 131-s − 6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 3·171-s + 173-s − 18·175-s + 179-s + 181-s + ⋯
L(s)  = 1  − 2.26·7-s − 9-s + 0.229·19-s + 3/5·25-s + 0.304·43-s + 13/7·49-s + 1.79·61-s + 2.26·63-s − 0.702·73-s + 81-s + 1.36·121-s + 0.0887·127-s + 0.0873·131-s − 0.520·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s − 0.229·171-s + 0.0760·173-s − 1.36·175-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(987696\)    =    \(2^{4} \cdot 3^{2} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(62.9763\)
Root analytic conductor: \(2.81704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 987696,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_2$ \( 1 + p T^{2} \)
19$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.11.a_ap
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.17.a_bd
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.29.a_be
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.a_abm
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.a_cg
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.41.a_abe
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.43.ac_dj
47$C_2^2$ \( 1 - 87 T^{2} + p^{2} T^{4} \) 2.47.a_adj
53$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.53.a_da
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.71.a_be
73$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.73.g_fz
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.89.a_acw
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−7.961129469380962181989895212878, −7.32305130052660723092311766345, −6.94425131620653819275823424387, −6.59760480537648271802059262796, −6.11070418133103761585213486013, −5.83615360060713533534812825033, −5.32506833441713424107768067043, −4.76444957510243367255024010339, −4.05768926443047187818056846955, −3.52186728479255860529351470583, −3.12901467819300418738126057534, −2.75614081137912399940820930647, −2.08483614218035674769636783112, −0.854302633003477795640210094843, 0, 0.854302633003477795640210094843, 2.08483614218035674769636783112, 2.75614081137912399940820930647, 3.12901467819300418738126057534, 3.52186728479255860529351470583, 4.05768926443047187818056846955, 4.76444957510243367255024010339, 5.32506833441713424107768067043, 5.83615360060713533534812825033, 6.11070418133103761585213486013, 6.59760480537648271802059262796, 6.94425131620653819275823424387, 7.32305130052660723092311766345, 7.961129469380962181989895212878

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