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Properties

Label 4-54e2-1.1-c1e2-0-2
Degree $4$
Conductor $2916$
Sign $1$
Analytic cond. $0.185926$
Root an. cond. $0.656652$
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  + 2-s − 2·7-s − 8-s − 3·11-s − 2·13-s − 2·14-s − 16-s + 6·17-s − 2·19-s − 3·22-s − 6·23-s + 5·25-s − 2·26-s + 6·29-s + 4·31-s + 6·34-s − 8·37-s − 2·38-s + 9·41-s + 43-s − 6·46-s − 6·47-s + 7·49-s + 5·50-s − 24·53-s + 2·56-s + 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.755·7-s − 0.353·8-s − 0.904·11-s − 0.554·13-s − 0.534·14-s − 1/4·16-s + 1.45·17-s − 0.458·19-s − 0.639·22-s − 1.25·23-s + 25-s − 0.392·26-s + 1.11·29-s + 0.718·31-s + 1.02·34-s − 1.31·37-s − 0.324·38-s + 1.40·41-s + 0.152·43-s − 0.884·46-s − 0.875·47-s + 49-s + 0.707·50-s − 3.29·53-s + 0.267·56-s + 0.787·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2916\)    =    \(2^{2} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(0.185926\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2916,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8715328686\)
\(L(\frac12)\) \(\approx\) \(0.8715328686\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
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$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_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.i_dm
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_bo
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2^2$ \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_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.i_d
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_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.m_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

−15.59861712213794375411483361128, −15.12688103669667857937205546939, −14.27482924604470976704252044001, −14.09692654269574911820876482947, −13.51017109175237931670563352488, −12.65616926729159557398117926597, −12.33645744571137792579080469536, −12.17511820172585237543301203042, −10.91361990419339911954793543618, −10.57427835288387395140515205717, −9.644156657279471627612142445110, −9.517129125896991291576106911816, −8.109217585919713043018362675136, −8.035203487704401873263663516929, −6.82058941932479782202029435549, −6.24346619468319563425589462519, −5.35381362714877193537420366400, −4.71102187921828010545904359307, −3.57176631322221135410380789907, −2.69831856592917915664558595168, 2.69831856592917915664558595168, 3.57176631322221135410380789907, 4.71102187921828010545904359307, 5.35381362714877193537420366400, 6.24346619468319563425589462519, 6.82058941932479782202029435549, 8.035203487704401873263663516929, 8.109217585919713043018362675136, 9.517129125896991291576106911816, 9.644156657279471627612142445110, 10.57427835288387395140515205717, 10.91361990419339911954793543618, 12.17511820172585237543301203042, 12.33645744571137792579080469536, 12.65616926729159557398117926597, 13.51017109175237931670563352488, 14.09692654269574911820876482947, 14.27482924604470976704252044001, 15.12688103669667857937205546939, 15.59861712213794375411483361128

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