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Properties

Label 4-8450e2-1.1-c1e2-0-5
Degree $4$
Conductor $71402500$
Sign $1$
Analytic cond. $4552.68$
Root an. cond. $8.21423$
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  − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 2·7-s − 4·8-s − 6·12-s + 4·14-s + 5·16-s − 6·17-s − 4·19-s + 4·21-s + 8·24-s + 2·27-s − 6·28-s + 12·29-s − 10·31-s − 6·32-s + 12·34-s − 8·37-s + 8·38-s − 8·42-s − 4·43-s − 6·47-s − 10·48-s + 49-s + 12·51-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.755·7-s − 1.41·8-s − 1.73·12-s + 1.06·14-s + 5/4·16-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 1.63·24-s + 0.384·27-s − 1.13·28-s + 2.22·29-s − 1.79·31-s − 1.06·32-s + 2.05·34-s − 1.31·37-s + 1.29·38-s − 1.23·42-s − 0.609·43-s − 0.875·47-s − 1.44·48-s + 1/7·49-s + 1.68·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71402500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71402500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71402500\)    =    \(2^{2} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(4552.68\)
Root analytic conductor: \(8.21423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71402500,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
13 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
7$D_{4}$ \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_d
11$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.11.a_t
17$D_{4}$ \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bo
19$D_{4}$ \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_p
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$D_{4}$ \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_de
31$D_{4}$ \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_dg
37$D_{4}$ \( 1 + 8 T + 63 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_cl
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_bq
47$D_{4}$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_dn
53$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.53.a_dz
59$D_{4}$ \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fm
61$D_{4}$ \( 1 - 10 T + 144 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_fo
67$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_be
71$D_{4}$ \( 1 + 24 T + 274 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.71.y_ko
73$D_{4}$ \( 1 + 2 T + 144 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fo
79$D_{4}$ \( 1 - 10 T + 180 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_gy
83$D_{4}$ \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_dw
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.89.as_jz
97$D_{4}$ \( 1 + 14 T + 240 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.97.o_jg
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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

−7.70159763153847020592154794589, −7.06319261241160112653114985900, −6.83729980532016153074889075816, −6.78117615017289791871843434470, −6.26505714252774725469879780777, −6.12751406285905291072317902981, −5.72796370606817822072151105382, −5.31989161292594133480300761325, −4.76421502225714321737705826706, −4.72468306914898848356492210801, −4.00514747075678077807763251912, −3.64807594523875912079169718548, −3.01095910504017499340444119535, −2.92986933555127470065663697791, −2.08367524765446660834179192254, −2.01810840342636357488797104811, −1.36356439308204282095743244696, −0.67308997981099210223202500171, 0, 0, 0.67308997981099210223202500171, 1.36356439308204282095743244696, 2.01810840342636357488797104811, 2.08367524765446660834179192254, 2.92986933555127470065663697791, 3.01095910504017499340444119535, 3.64807594523875912079169718548, 4.00514747075678077807763251912, 4.72468306914898848356492210801, 4.76421502225714321737705826706, 5.31989161292594133480300761325, 5.72796370606817822072151105382, 6.12751406285905291072317902981, 6.26505714252774725469879780777, 6.78117615017289791871843434470, 6.83729980532016153074889075816, 7.06319261241160112653114985900, 7.70159763153847020592154794589

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