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Properties

Label 4-2940e2-1.1-c1e2-0-22
Degree $4$
Conductor $8643600$
Sign $1$
Analytic cond. $551.123$
Root an. cond. $4.84520$
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  − 2·3-s − 2·5-s + 3·9-s + 2·11-s + 4·15-s − 2·17-s − 6·19-s − 2·23-s + 3·25-s − 4·27-s + 10·29-s − 2·31-s − 4·33-s − 4·37-s + 6·41-s + 4·43-s − 6·45-s − 12·47-s + 4·51-s − 4·53-s − 4·55-s + 12·57-s − 6·59-s − 16·61-s − 4·67-s + 4·69-s + 22·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 9-s + 0.603·11-s + 1.03·15-s − 0.485·17-s − 1.37·19-s − 0.417·23-s + 3/5·25-s − 0.769·27-s + 1.85·29-s − 0.359·31-s − 0.696·33-s − 0.657·37-s + 0.937·41-s + 0.609·43-s − 0.894·45-s − 1.75·47-s + 0.560·51-s − 0.549·53-s − 0.539·55-s + 1.58·57-s − 0.781·59-s − 2.04·61-s − 0.488·67-s + 0.481·69-s + 2.61·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8643600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8643600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(551.123\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 8643600,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good11$D_{4}$ \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_q
13$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.13.a_t
17$D_{4}$ \( 1 + 2 T + 28 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_bc
19$D_{4}$ \( 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_t
23$D_{4}$ \( 1 + 2 T + 40 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bo
29$D_{4}$ \( 1 - 10 T + 76 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.29.ak_cy
31$D_{4}$ \( 1 + 2 T + 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.31.c_bj
37$D_{4}$ \( 1 + 4 T + 71 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_ct
41$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_bc
43$D_{4}$ \( 1 - 4 T + 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_bb
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$D_{4}$ \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_de
59$D_{4}$ \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_cm
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$D_{4}$ \( 1 + 4 T + 131 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_fb
71$D_{4}$ \( 1 - 22 T + 256 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.71.aw_jw
73$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.73.a_abd
79$D_{4}$ \( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_fj
83$D_{4}$ \( 1 - 6 T + 112 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_ei
89$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_e
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−8.336898924447865716572818098314, −8.307462494790344301334692836849, −7.75112764928530717455200414400, −7.49332816366976784694514194768, −6.87246767725945517785513397008, −6.47866242269807619472835415173, −6.43036861165827495316228777643, −6.14308722438061185073775956683, −5.28651346875197164790812005111, −5.17164068998328957742613280428, −4.55429118265446667781241671394, −4.37905654834098986940840298752, −3.81018106522401852077228907246, −3.66199299500958599632960753517, −2.63450416022290959496984609206, −2.56556629290785875711644507677, −1.37069402499890836699197827376, −1.34341653690519261556267413256, 0, 0, 1.34341653690519261556267413256, 1.37069402499890836699197827376, 2.56556629290785875711644507677, 2.63450416022290959496984609206, 3.66199299500958599632960753517, 3.81018106522401852077228907246, 4.37905654834098986940840298752, 4.55429118265446667781241671394, 5.17164068998328957742613280428, 5.28651346875197164790812005111, 6.14308722438061185073775956683, 6.43036861165827495316228777643, 6.47866242269807619472835415173, 6.87246767725945517785513397008, 7.49332816366976784694514194768, 7.75112764928530717455200414400, 8.307462494790344301334692836849, 8.336898924447865716572818098314

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