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Properties

Label 4-1512e2-1.1-c1e2-0-29
Degree $4$
Conductor $2286144$
Sign $1$
Analytic cond. $145.766$
Root an. cond. $3.47467$
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  − 2·7-s − 4·11-s − 2·19-s − 8·23-s − 3·25-s − 12·29-s − 6·31-s + 6·37-s − 12·41-s − 4·43-s − 12·47-s + 3·49-s − 4·59-s + 8·61-s + 8·67-s + 12·73-s + 8·77-s − 12·79-s − 20·83-s + 12·89-s + 16·97-s − 8·101-s + 2·103-s − 16·107-s − 2·109-s − 8·113-s − 3·121-s + ⋯
L(s)  = 1  − 0.755·7-s − 1.20·11-s − 0.458·19-s − 1.66·23-s − 3/5·25-s − 2.22·29-s − 1.07·31-s + 0.986·37-s − 1.87·41-s − 0.609·43-s − 1.75·47-s + 3/7·49-s − 0.520·59-s + 1.02·61-s + 0.977·67-s + 1.40·73-s + 0.911·77-s − 1.35·79-s − 2.19·83-s + 1.27·89-s + 1.62·97-s − 0.796·101-s + 0.197·103-s − 1.54·107-s − 0.191·109-s − 0.752·113-s − 0.272·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(145.766\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
11$D_{4}$ \( 1 + 4 T + 19 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_t
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$D_{4}$ \( 1 + 8 T + 55 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_cd
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2^2$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_br
37$D_{4}$ \( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_cd
41$D_{4}$ \( 1 + 12 T + 111 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_eh
43$D_{4}$ \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_ck
47$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_dy
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$D_{4}$ \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_dq
61$D_{4}$ \( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_ba
67$D_{4}$ \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_es
71$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.71.a_abh
73$D_{4}$ \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_fy
79$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_gk
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.83.u_kg
89$D_{4}$ \( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_fv
97$D_{4}$ \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_fq
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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

−9.542654652216230959521704349193, −8.843937288087059936676586379472, −8.346933451372506979963474538918, −8.163135103551902999998789279613, −7.53166886724824844002599940182, −7.51836469526385514164086456315, −6.65838950182135738678503470550, −6.58458565192231041921810364338, −5.90998775372738851706530556547, −5.58247863228340094066682251878, −5.23171088538103340208234197605, −4.73215326334465849502908556099, −3.94257712045845845227713106711, −3.76830950206860204984510345036, −3.27204007459803713190804078611, −2.56955837779224846930202337162, −2.05297410657727230163455175933, −1.59467472235857577988802052659, 0, 0, 1.59467472235857577988802052659, 2.05297410657727230163455175933, 2.56955837779224846930202337162, 3.27204007459803713190804078611, 3.76830950206860204984510345036, 3.94257712045845845227713106711, 4.73215326334465849502908556099, 5.23171088538103340208234197605, 5.58247863228340094066682251878, 5.90998775372738851706530556547, 6.58458565192231041921810364338, 6.65838950182135738678503470550, 7.51836469526385514164086456315, 7.53166886724824844002599940182, 8.163135103551902999998789279613, 8.346933451372506979963474538918, 8.843937288087059936676586379472, 9.542654652216230959521704349193

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