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Properties

Label 4-744e2-1.1-c1e2-0-26
Degree $4$
Conductor $553536$
Sign $1$
Analytic cond. $35.2939$
Root an. cond. $2.43738$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s + 2·11-s − 13-s + 4·15-s − 7·19-s − 12·23-s + 5·25-s + 27-s − 16·29-s + 4·31-s − 2·33-s − 7·37-s + 39-s − 10·41-s − 7·43-s + 16·47-s + 7·49-s − 2·53-s − 8·55-s + 7·57-s − 12·61-s + 4·65-s − 12·67-s + 12·69-s − 6·71-s + 7·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 0.603·11-s − 0.277·13-s + 1.03·15-s − 1.60·19-s − 2.50·23-s + 25-s + 0.192·27-s − 2.97·29-s + 0.718·31-s − 0.348·33-s − 1.15·37-s + 0.160·39-s − 1.56·41-s − 1.06·43-s + 2.33·47-s + 49-s − 0.274·53-s − 1.07·55-s + 0.927·57-s − 1.53·61-s + 0.496·65-s − 1.46·67-s + 1.44·69-s − 0.712·71-s + 0.819·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 553536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 553536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(553536\)    =    \(2^{6} \cdot 3^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(35.2939\)
Root analytic conductor: \(2.43738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 553536,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
31$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_l
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_am
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.h_be
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.29.q_es
37$C_2^2$ \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.37.h_m
41$C_2^2$ \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_ch
43$C_2^2$ \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.43.h_g
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.53.c_abx
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_cz
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.ah_ay
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.79.a_adb
83$C_2^2$ \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_acp
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.97.o_jj
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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

−10.32165801844977326120532863574, −9.815742984975538980958403683791, −9.066706901098375683454219171018, −9.002128196817832553354133638352, −8.199383024586163812051703172797, −8.066747505713466329000281803737, −7.59858902260661772856832001490, −7.14115856533176818754357129917, −6.71686042770442582929074003690, −6.13050784041457670646743132343, −5.73787570426129041763139246492, −5.25644589924569892567826147377, −4.28814214517130195726052654429, −4.26656324138919095511737106704, −3.77418175541079537806180790539, −3.30796038011177653728720803610, −2.20174789807897169143214863835, −1.69359561366068207212931743756, 0, 0, 1.69359561366068207212931743756, 2.20174789807897169143214863835, 3.30796038011177653728720803610, 3.77418175541079537806180790539, 4.26656324138919095511737106704, 4.28814214517130195726052654429, 5.25644589924569892567826147377, 5.73787570426129041763139246492, 6.13050784041457670646743132343, 6.71686042770442582929074003690, 7.14115856533176818754357129917, 7.59858902260661772856832001490, 8.066747505713466329000281803737, 8.199383024586163812051703172797, 9.002128196817832553354133638352, 9.066706901098375683454219171018, 9.815742984975538980958403683791, 10.32165801844977326120532863574

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