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Properties

Label 4-7098e2-1.1-c1e2-0-11
Degree $4$
Conductor $50381604$
Sign $1$
Analytic cond. $3212.37$
Root an. cond. $7.52846$
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 + 5-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s − 2·10-s − 5·11-s − 6·12-s + 4·14-s − 2·15-s + 5·16-s + 17-s − 6·18-s − 5·19-s + 3·20-s + 4·21-s + 10·22-s + 3·23-s + 8·24-s + 25-s − 4·27-s − 6·28-s − 29-s + 4·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.447·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 0.632·10-s − 1.50·11-s − 1.73·12-s + 1.06·14-s − 0.516·15-s + 5/4·16-s + 0.242·17-s − 1.41·18-s − 1.14·19-s + 0.670·20-s + 0.872·21-s + 2.13·22-s + 0.625·23-s + 1.63·24-s + 1/5·25-s − 0.769·27-s − 1.13·28-s − 0.185·29-s + 0.730·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50381604\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(3212.37\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 50381604,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good5$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \) 2.5.ab_a
11$D_{4}$ \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_s
17$D_{4}$ \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_y
19$D_{4}$ \( 1 + 5 T + 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_bi
23$D_{4}$ \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.23.ad_bm
29$D_{4}$ \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_bw
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$D_{4}$ \( 1 + 15 T + 120 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.37.p_eq
41$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_by
43$D_{4}$ \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_ag
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_abq
61$D_{4}$ \( 1 - 9 T + 132 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.61.aj_fc
67$D_{4}$ \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_dq
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$D_{4}$ \( 1 - 9 T + 156 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.73.aj_ga
79$D_{4}$ \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_ew
83$D_{4}$ \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.83.ao_gs
89$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.89.i_be
97$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_bu
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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

−7.62133593856147588864974082567, −7.57976194336651102367932624854, −6.83229182550392748881287348510, −6.79553145719867227607933891596, −6.63963338988380843288107994114, −6.08719747822843918064729502805, −5.63247540109401212554825950848, −5.44779964744874416962042712709, −5.15011245663189714527164051163, −4.79799817969987356744483761555, −3.96679014257129331235462057862, −3.83402339354698531677533686262, −3.27077128899533600048698914975, −2.62695951803637829494216707010, −2.30762536580100188551238448618, −2.11247323564436704233612143946, −1.22100104579519576297249934945, −0.914114483621225028889625908786, 0, 0, 0.914114483621225028889625908786, 1.22100104579519576297249934945, 2.11247323564436704233612143946, 2.30762536580100188551238448618, 2.62695951803637829494216707010, 3.27077128899533600048698914975, 3.83402339354698531677533686262, 3.96679014257129331235462057862, 4.79799817969987356744483761555, 5.15011245663189714527164051163, 5.44779964744874416962042712709, 5.63247540109401212554825950848, 6.08719747822843918064729502805, 6.63963338988380843288107994114, 6.79553145719867227607933891596, 6.83229182550392748881287348510, 7.57976194336651102367932624854, 7.62133593856147588864974082567

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