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Properties

Label 4-1792e2-1.1-c1e2-0-18
Degree $4$
Conductor $3211264$
Sign $1$
Analytic cond. $204.752$
Root an. cond. $3.78274$
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  − 2·3-s + 2·5-s + 2·7-s + 2·9-s − 8·11-s − 2·13-s − 4·15-s − 4·17-s + 2·19-s − 4·21-s + 4·23-s − 2·25-s − 6·27-s − 12·29-s + 16·33-s + 4·35-s − 4·37-s + 4·39-s − 4·41-s + 4·45-s − 8·47-s + 3·49-s + 8·51-s − 16·55-s − 4·57-s − 14·59-s + 2·61-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 0.755·7-s + 2/3·9-s − 2.41·11-s − 0.554·13-s − 1.03·15-s − 0.970·17-s + 0.458·19-s − 0.872·21-s + 0.834·23-s − 2/5·25-s − 1.15·27-s − 2.22·29-s + 2.78·33-s + 0.676·35-s − 0.657·37-s + 0.640·39-s − 0.624·41-s + 0.596·45-s − 1.16·47-s + 3/7·49-s + 1.12·51-s − 2.15·55-s − 0.529·57-s − 1.82·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3211264\)    =    \(2^{16} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(204.752\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3211264,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_c
5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_g
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_w
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$D_{4}$ \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_bi
23$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_be
29$C_4$ \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.29.m_cw
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_cg
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.43.a_g
47$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_be
53$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.53.a_ba
59$D_{4}$ \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.59.o_gg
61$D_{4}$ \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_eo
67$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_fu
71$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_da
73$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_dy
79$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_dq
83$D_{4}$ \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.83.o_ic
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.89.ae_ha
97$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_eo
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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

−9.350192855167780744727517324046, −8.639317914738830253527066503883, −8.226918823380681324938839163272, −7.77185370798865268721872065279, −7.53112219730028550957823463418, −7.15685392420471898896199678347, −6.63506693508790279560739365532, −6.18997609759406507955268760571, −5.51917579658935556738798755969, −5.48289574234310409620671623617, −5.23061835018876204019277200896, −4.79589178773493400445909396160, −4.29239920115117363035585128496, −3.63971843644313591057156594938, −2.91077978729400868052941694003, −2.47547224727555636269706555927, −1.86105559406579022162662483233, −1.51218650745078128786791197455, 0, 0, 1.51218650745078128786791197455, 1.86105559406579022162662483233, 2.47547224727555636269706555927, 2.91077978729400868052941694003, 3.63971843644313591057156594938, 4.29239920115117363035585128496, 4.79589178773493400445909396160, 5.23061835018876204019277200896, 5.48289574234310409620671623617, 5.51917579658935556738798755969, 6.18997609759406507955268760571, 6.63506693508790279560739365532, 7.15685392420471898896199678347, 7.53112219730028550957823463418, 7.77185370798865268721872065279, 8.226918823380681324938839163272, 8.639317914738830253527066503883, 9.350192855167780744727517324046

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