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Properties

Label 4-24e4-1.1-c1e2-0-47
Degree $4$
Conductor $331776$
Sign $-1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
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  − 4·17-s + 2·25-s − 20·41-s − 6·49-s + 4·73-s + 12·89-s − 28·97-s − 4·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 0.970·17-s + 2/5·25-s − 3.12·41-s − 6/7·49-s + 0.468·73-s + 1.27·89-s − 2.84·97-s − 0.376·113-s − 0.545·121-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 − 0.461·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(21.1543\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ -1)\)

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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.29.a_aby
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.37.a_acw
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.47.a_ck
53$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.53.a_adu
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.61.a_g
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.71.a_eg
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.79.a_di
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−8.659419060185556688652600447029, −8.120861377745114095262652060409, −7.66290031684715219109380088279, −7.02040768550923387363137955424, −6.61907139542909577530501088742, −6.39408113550237778948715224326, −5.58017056580972558320541358952, −5.10220642781078545886602696158, −4.71955000823213590895002857444, −4.03739737234269307525350964101, −3.47817577765181009119187651197, −2.86813266448153326435766858426, −2.11425319901079415089435207612, −1.39661700422223217169569454597, 0, 1.39661700422223217169569454597, 2.11425319901079415089435207612, 2.86813266448153326435766858426, 3.47817577765181009119187651197, 4.03739737234269307525350964101, 4.71955000823213590895002857444, 5.10220642781078545886602696158, 5.58017056580972558320541358952, 6.39408113550237778948715224326, 6.61907139542909577530501088742, 7.02040768550923387363137955424, 7.66290031684715219109380088279, 8.120861377745114095262652060409, 8.659419060185556688652600447029

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