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Properties

Label 4-864e2-1.1-c1e2-0-33
Degree $4$
Conductor $746496$
Sign $-1$
Analytic cond. $47.5972$
Root an. cond. $2.62660$
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  + 2·7-s − 4·13-s − 25-s − 10·31-s − 4·43-s + 5·49-s + 20·67-s − 6·73-s − 8·91-s − 10·97-s − 8·103-s − 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 0.755·7-s − 1.10·13-s − 1/5·25-s − 1.79·31-s − 0.609·43-s + 5/7·49-s + 2.44·67-s − 0.702·73-s − 0.838·91-s − 1.01·97-s − 0.788·103-s − 0.454·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s − 0.151·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(746496\)    =    \(2^{10} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(47.5972\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 746496,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.ac_ab
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.e_ba
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.k_ct
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.41.a_ak
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.e_ba
47$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.47.a_k
53$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \) 2.53.a_bh
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.67.au_ja
71$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.71.a_be
73$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.73.g_l
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2^2$ \( 1 - 43 T^{2} + p^{2} T^{4} \) 2.83.a_abr
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.89.a_be
97$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.97.k_cx
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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

−8.041764858913999176782613181687, −7.63121832990769535402464390015, −7.15628710462374356290328793593, −6.90861711679362434497792107114, −6.26197369189457851526778279155, −5.68824214952961854645032940202, −5.17052616635325985880632649184, −5.03093621029957419457886585318, −4.27815657140639024747881011562, −3.85810000901155997239535168085, −3.24582912488533076545035269769, −2.45636316817433222078530078293, −2.03980379822483563768201267843, −1.24625155279638920608582671287, 0, 1.24625155279638920608582671287, 2.03980379822483563768201267843, 2.45636316817433222078530078293, 3.24582912488533076545035269769, 3.85810000901155997239535168085, 4.27815657140639024747881011562, 5.03093621029957419457886585318, 5.17052616635325985880632649184, 5.68824214952961854645032940202, 6.26197369189457851526778279155, 6.90861711679362434497792107114, 7.15628710462374356290328793593, 7.63121832990769535402464390015, 8.041764858913999176782613181687

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