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Properties

Label 4-24e4-1.1-c1e2-0-24
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $21.1543$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2·7-s + 6·9-s − 3·11-s + 2·13-s − 6·17-s + 2·19-s − 6·21-s + 6·23-s + 5·25-s + 9·27-s + 6·29-s + 4·31-s − 9·33-s + 8·37-s + 6·39-s − 9·41-s − 43-s + 6·47-s + 7·49-s − 18·51-s − 24·53-s + 6·57-s + 3·59-s + 8·61-s − 12·63-s + 5·67-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.755·7-s + 2·9-s − 0.904·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s − 1.30·21-s + 1.25·23-s + 25-s + 1.73·27-s + 1.11·29-s + 0.718·31-s − 1.56·33-s + 1.31·37-s + 0.960·39-s − 1.40·41-s − 0.152·43-s + 0.875·47-s + 49-s − 2.52·51-s − 3.29·53-s + 0.794·57-s + 0.390·59-s + 1.02·61-s − 1.51·63-s + 0.610·67-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.190221062\)
\(L(\frac12)\) \(\approx\) \(3.190221062\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.ac_aj
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_al
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_aby
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_d
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.af_abq
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.73.aw_kh
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.f_acu
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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

−10.72305383898545842725791060104, −10.53451323853155045925429306928, −9.883836769546784319216918455702, −9.533581995551175569466065716540, −9.178052544955769210267289398166, −8.654396715069845586623535213226, −8.459661689076367849449258784608, −7.960549983500854019345144533333, −7.47059502115549113841695363803, −6.97653707379361751422972252768, −6.48424228187937617705928151915, −6.23998793287684594489211087629, −5.16673678918970595668852023714, −4.79436426235201640162779404909, −4.27671346041948067499332489713, −3.51590222255242477545000646547, −3.04339839537422185967313347664, −2.70223626288530671589686135578, −2.05369672378010761092922360868, −0.963878146994308968468362047498, 0.963878146994308968468362047498, 2.05369672378010761092922360868, 2.70223626288530671589686135578, 3.04339839537422185967313347664, 3.51590222255242477545000646547, 4.27671346041948067499332489713, 4.79436426235201640162779404909, 5.16673678918970595668852023714, 6.23998793287684594489211087629, 6.48424228187937617705928151915, 6.97653707379361751422972252768, 7.47059502115549113841695363803, 7.960549983500854019345144533333, 8.459661689076367849449258784608, 8.654396715069845586623535213226, 9.178052544955769210267289398166, 9.533581995551175569466065716540, 9.883836769546784319216918455702, 10.53451323853155045925429306928, 10.72305383898545842725791060104

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