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Properties

Label 4-1456e2-1.1-c1e2-0-28
Degree $4$
Conductor $2119936$
Sign $1$
Analytic cond. $135.168$
Root an. cond. $3.40972$
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  − 2·5-s + 2·7-s − 8·11-s − 2·13-s − 6·19-s + 2·23-s − 25-s − 2·29-s − 2·31-s − 4·35-s − 12·37-s + 12·41-s − 2·43-s − 2·47-s + 3·49-s − 6·53-s + 16·55-s − 28·59-s − 4·61-s + 4·65-s + 4·67-s − 20·71-s − 2·73-s − 16·77-s + 14·79-s − 9·81-s − 6·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s − 2.41·11-s − 0.554·13-s − 1.37·19-s + 0.417·23-s − 1/5·25-s − 0.371·29-s − 0.359·31-s − 0.676·35-s − 1.97·37-s + 1.87·41-s − 0.304·43-s − 0.291·47-s + 3/7·49-s − 0.824·53-s + 2.15·55-s − 3.64·59-s − 0.512·61-s + 0.496·65-s + 0.488·67-s − 2.37·71-s − 0.234·73-s − 1.82·77-s + 1.57·79-s − 81-s − 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2119936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2119936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2119936\)    =    \(2^{8} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(135.168\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2119936,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
5$D_{4}$ \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_f
11$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.11.i_bg
17$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.17.a_bc
19$D_{4}$ \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bp
23$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_x
29$D_{4}$ \( 1 + 2 T + 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_bj
31$D_{4}$ \( 1 + 2 T + 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_cf
37$D_{4}$ \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_ea
41$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_dq
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$D_{4}$ \( 1 + 2 T + 89 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_dl
53$D_{4}$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_dn
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.59.bc_mc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$D_{4}$ \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_ek
71$D_{4}$ \( 1 + 20 T + 236 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.71.u_jc
73$D_{4}$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_ad
79$D_{4}$ \( 1 - 14 T + 183 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.79.ao_hb
83$D_{4}$ \( 1 + 6 T + 121 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.83.g_er
89$D_{4}$ \( 1 + 18 T + 253 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_jt
97$D_{4}$ \( 1 + 26 T + 357 T^{2} + 26 p T^{3} + p^{2} T^{4} \) 2.97.ba_nt
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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

−9.200686159916403648839369721504, −8.845371129730723967554545584376, −8.355341703671758276282817715156, −8.132387587166883071479504580285, −7.67409235701984372739636576867, −7.34583102010641700425176161061, −7.26586601235750076452220794070, −6.44719784554833115886527055218, −5.85937170281570623192006023466, −5.64211993005444421707029681347, −4.97825894742776151509125159063, −4.67039898857974993423596354831, −4.42276674001154711894334487428, −3.73717950064701044936674308446, −3.04738977331923578819196461222, −2.76519383144708211099522139967, −2.06816494975524882831877428014, −1.53911984482777626896834633845, 0, 0, 1.53911984482777626896834633845, 2.06816494975524882831877428014, 2.76519383144708211099522139967, 3.04738977331923578819196461222, 3.73717950064701044936674308446, 4.42276674001154711894334487428, 4.67039898857974993423596354831, 4.97825894742776151509125159063, 5.64211993005444421707029681347, 5.85937170281570623192006023466, 6.44719784554833115886527055218, 7.26586601235750076452220794070, 7.34583102010641700425176161061, 7.67409235701984372739636576867, 8.132387587166883071479504580285, 8.355341703671758276282817715156, 8.845371129730723967554545584376, 9.200686159916403648839369721504

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