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Properties

Label 4-864e2-1.1-c1e2-0-32
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  + 3·11-s − 6·17-s − 4·19-s − 25-s + 15·41-s − 7·43-s − 49-s − 3·59-s − 13·67-s − 2·73-s − 18·83-s + 12·89-s − 11·97-s + 12·107-s + 18·113-s − 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + ⋯
L(s)  = 1  + 0.904·11-s − 1.45·17-s − 0.917·19-s − 1/5·25-s + 2.34·41-s − 1.06·43-s − 1/7·49-s − 0.390·59-s − 1.58·67-s − 0.234·73-s − 1.97·83-s + 1.27·89-s − 1.11·97-s + 1.16·107-s + 1.69·113-s − 1.18·121-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 − 0.0769·169-s + 0.0760·173-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$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.7.a_b
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ad_w
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.e_bh
23$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.23.a_b
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.31.a_b
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.37.a_w
41$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.41.ap_fg
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.h_da
47$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \) 2.47.a_bx
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.d_eo
61$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.61.a_z
67$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.n_gs
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.c_d
79$C_2^2$ \( 1 - 95 T^{2} + p^{2} T^{4} \) 2.79.a_adr
83$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.s_id
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.l_gm
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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

−8.139879999706650607630818426549, −7.53698372169676512967766055442, −7.19100591143755850098122194963, −6.65843299115384187384283847011, −6.25662047054462198451966054246, −5.98182840297544678918200700265, −5.34642511850129884343260547644, −4.54456649235047191034996686040, −4.42869614756654881351689080083, −3.90333999023696151080314717295, −3.22689153403607592314696002734, −2.53359883817303697993227419212, −1.99627903342694064370485302700, −1.22399319577796990751985147301, 0, 1.22399319577796990751985147301, 1.99627903342694064370485302700, 2.53359883817303697993227419212, 3.22689153403607592314696002734, 3.90333999023696151080314717295, 4.42869614756654881351689080083, 4.54456649235047191034996686040, 5.34642511850129884343260547644, 5.98182840297544678918200700265, 6.25662047054462198451966054246, 6.65843299115384187384283847011, 7.19100591143755850098122194963, 7.53698372169676512967766055442, 8.139879999706650607630818426549

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