↑

Properties

Label 4-1512e2-1.1-c1e2-0-26
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 12·11-s + 4·16-s − 6·17-s + 4·19-s − 25-s + 6·41-s − 2·43-s + 24·44-s + 49-s − 18·59-s − 8·64-s − 8·67-s + 12·68-s + 4·73-s − 8·76-s − 6·83-s − 12·89-s − 20·97-s + 2·100-s − 12·107-s + 24·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4-s − 3.61·11-s + 16-s − 1.45·17-s + 0.917·19-s − 1/5·25-s + 0.937·41-s − 0.304·43-s + 3.61·44-s + 1/7·49-s − 2.34·59-s − 64-s − 0.977·67-s + 1.45·68-s + 0.468·73-s − 0.917·76-s − 0.658·83-s − 1.27·89-s − 2.03·97-s + 1/5·100-s − 1.16·107-s + 2.25·113-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.a_z
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.41.ag_dn
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.59.s_hr
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.a_gb
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.83.g_gt
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.29298251764697241533648179098, −7.15805925524584729425198877495, −6.22235231089039947188465974061, −5.89880361188496118864247496902, −5.37953669731434672726098965475, −5.18288713324048865561083127649, −4.57429112780059186549971965551, −4.52243663618215258937754394263, −3.67044930538295279435088265727, −3.05701292571172651199468963397, −2.66361329360886394934731917597, −2.28013168161255667505063368593, −1.28972514507845589157072210561, 0, 0, 1.28972514507845589157072210561, 2.28013168161255667505063368593, 2.66361329360886394934731917597, 3.05701292571172651199468963397, 3.67044930538295279435088265727, 4.52243663618215258937754394263, 4.57429112780059186549971965551, 5.18288713324048865561083127649, 5.37953669731434672726098965475, 5.89880361188496118864247496902, 6.22235231089039947188465974061, 7.15805925524584729425198877495, 7.29298251764697241533648179098

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