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Properties

Label 4-1184e2-1.1-c1e2-0-9
Degree $4$
Conductor $1401856$
Sign $1$
Analytic cond. $89.3835$
Root an. cond. $3.07478$
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  − 6·3-s − 6·7-s + 21·9-s + 6·11-s + 36·21-s + 10·25-s − 54·27-s − 36·33-s − 2·37-s − 14·41-s − 6·47-s + 13·49-s − 22·53-s − 126·63-s − 24·67-s + 6·71-s − 14·73-s − 60·75-s − 36·77-s + 108·81-s − 18·83-s + 126·99-s − 22·101-s + 24·107-s + 12·111-s + 5·121-s + 84·123-s + ⋯
L(s)  = 1  − 3.46·3-s − 2.26·7-s + 7·9-s + 1.80·11-s + 7.85·21-s + 2·25-s − 10.3·27-s − 6.26·33-s − 0.328·37-s − 2.18·41-s − 0.875·47-s + 13/7·49-s − 3.02·53-s − 15.8·63-s − 2.93·67-s + 0.712·71-s − 1.63·73-s − 6.92·75-s − 4.10·77-s + 12·81-s − 1.97·83-s + 12.6·99-s − 2.18·101-s + 2.32·107-s + 1.13·111-s + 5/11·121-s + 7.57·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1401856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1401856\)    =    \(2^{10} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(89.3835\)
Root analytic conductor: \(3.07478\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1401856,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
37$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T + p T^{2} )^{2} \) 2.3.g_p
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.41.o_fb
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.53.w_it
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.59.a_as
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.61.a_adi
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.71.ag_fv
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \) 2.79.a_aes
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.83.s_jn
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.89.a_abi
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \) 2.97.a_agc
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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

−9.717098620071289271482367988573, −9.476453010398267225649005615815, −8.911765216794801910154404843509, −8.526835917607936880576998432277, −7.42495547877824549447774877256, −7.15790908594484786264171609282, −6.65295818064336061451589585980, −6.48215044793463326219237435036, −6.30325402447759243699073845885, −6.03951397522205891602003801067, −5.33234874297983353803064831620, −5.03230299323480301255415095935, −4.47812985944761101242708587188, −4.18178916827378162770762962432, −3.22287706565999957455946503338, −3.20865092820049952162660663364, −1.43118531660884195734322183492, −1.28642240988567014858029642864, 0, 0, 1.28642240988567014858029642864, 1.43118531660884195734322183492, 3.20865092820049952162660663364, 3.22287706565999957455946503338, 4.18178916827378162770762962432, 4.47812985944761101242708587188, 5.03230299323480301255415095935, 5.33234874297983353803064831620, 6.03951397522205891602003801067, 6.30325402447759243699073845885, 6.48215044793463326219237435036, 6.65295818064336061451589585980, 7.15790908594484786264171609282, 7.42495547877824549447774877256, 8.526835917607936880576998432277, 8.911765216794801910154404843509, 9.476453010398267225649005615815, 9.717098620071289271482367988573

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