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Properties

Label 4-2178e2-1.1-c1e2-0-36
Degree $4$
Conductor $4743684$
Sign $1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s − 6·7-s + 4·8-s − 6·13-s − 12·14-s + 5·16-s − 6·19-s + 6·23-s − 7·25-s − 12·26-s − 18·28-s − 6·29-s − 10·31-s + 6·32-s − 4·37-s − 12·38-s − 12·41-s + 12·46-s + 6·47-s + 16·49-s − 14·50-s − 18·52-s − 12·53-s − 24·56-s − 12·58-s − 12·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 2.26·7-s + 1.41·8-s − 1.66·13-s − 3.20·14-s + 5/4·16-s − 1.37·19-s + 1.25·23-s − 7/5·25-s − 2.35·26-s − 3.40·28-s − 1.11·29-s − 1.79·31-s + 1.06·32-s − 0.657·37-s − 1.94·38-s − 1.87·41-s + 1.76·46-s + 0.875·47-s + 16/7·49-s − 1.97·50-s − 2.49·52-s − 1.64·53-s − 3.20·56-s − 1.57·58-s − 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4743684\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(302.461\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
11 \( 1 \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$D_{4}$ \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_u
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.13.g_bj
17$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.17.a_h
19$D_{4}$ \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bs
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_ca
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$D_{4}$ \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_ci
37$D_{4}$ \( 1 + 4 T + 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_bz
41$D_{4}$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dn
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$D_{4}$ \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_cy
53$D_{4}$ \( 1 + 12 T + 115 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_el
59$D_{4}$ \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fm
61$D_{4}$ \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_fq
67$D_{4}$ \( 1 + 10 T + 132 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_fc
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_gk
73$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_fe
79$D_{4}$ \( 1 - 6 T + 164 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_gi
83$D_{4}$ \( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_fs
89$D_{4}$ \( 1 + 6 T + 175 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_gt
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
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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

−9.027219211629765449870574725073, −8.590027358934258984720025276059, −7.64501864794063013815953914908, −7.64107503646323270620249412976, −7.20487945566184512677378007240, −6.78772293748463371709781686832, −6.33966068652661967401586500006, −6.26454186299328813581146942270, −5.60517856043032190864113790163, −5.40272703936423579837973669233, −4.65239771122132418890426332691, −4.61500109267265952492823062272, −3.66633512567868361237070146886, −3.66473770872035042467056476711, −3.06419575855914689132187049858, −2.80271186854417334635065665712, −1.98509485485938539927827423007, −1.78253948958468122601246220165, 0, 0, 1.78253948958468122601246220165, 1.98509485485938539927827423007, 2.80271186854417334635065665712, 3.06419575855914689132187049858, 3.66473770872035042467056476711, 3.66633512567868361237070146886, 4.61500109267265952492823062272, 4.65239771122132418890426332691, 5.40272703936423579837973669233, 5.60517856043032190864113790163, 6.26454186299328813581146942270, 6.33966068652661967401586500006, 6.78772293748463371709781686832, 7.20487945566184512677378007240, 7.64107503646323270620249412976, 7.64501864794063013815953914908, 8.590027358934258984720025276059, 9.027219211629765449870574725073

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