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Properties

Label 4-1850e2-1.1-c1e2-0-19
Degree $4$
Conductor $3422500$
Sign $1$
Analytic cond. $218.221$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

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 + 4·8-s + 3·9-s + 2·11-s + 6·12-s + 4·13-s + 5·16-s + 2·17-s + 6·18-s − 10·19-s + 4·22-s + 4·23-s + 8·24-s + 8·26-s + 10·27-s + 8·29-s + 4·31-s + 6·32-s + 4·33-s + 4·34-s + 9·36-s − 2·37-s − 20·38-s + 8·39-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s + 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s + 1.10·13-s + 5/4·16-s + 0.485·17-s + 1.41·18-s − 2.29·19-s + 0.852·22-s + 0.834·23-s + 1.63·24-s + 1.56·26-s + 1.92·27-s + 1.48·29-s + 0.718·31-s + 1.06·32-s + 0.696·33-s + 0.685·34-s + 3/2·36-s − 0.328·37-s − 3.24·38-s + 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3422500\)    =    \(2^{2} \cdot 5^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(218.221\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(12.23129348\)
\(L(\frac12)\) \(\approx\) \(12.23129348\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_b
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.7.a_i
11$D_{4}$ \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_r
13$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_y
17$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_bd
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.19.k_cl
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_by
31$D_{4}$ \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_ci
41$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.41.ac_df
43$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_du
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.47.a_ac
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.59.e_es
61$D_{4}$ \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_fc
67$D_{4}$ \( 1 - 14 T + 177 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_gv
71$D_{4}$ \( 1 + 20 T + 236 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.71.u_jc
73$D_{4}$ \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.73.ag_ch
79$D_{4}$ \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_co
83$D_{4}$ \( 1 - 2 T + 161 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_gf
89$D_{4}$ \( 1 - 14 T + 173 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.89.ao_gr
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−9.185075631281304424842418069504, −8.942923879399788595071668107048, −8.561939574147500571790315438538, −8.396879899200597943631683923191, −7.80974681604262524886903360551, −7.53204207622829071181453656909, −6.75553309758946923431375030008, −6.56364957581995654575158709468, −6.38428722623152595777496316589, −6.00293497616628195120104017780, −5.13932141230260486730540470487, −4.82604220680341334907637978048, −4.50299375581603978348995533716, −3.99537293944878661435748904089, −3.54110406293956700135370985156, −3.25827034717114569518100019075, −2.65893092684107510662709321005, −2.26511061878540504975212372415, −1.56276024939701118332757226885, −1.03133138935126130223078502192, 1.03133138935126130223078502192, 1.56276024939701118332757226885, 2.26511061878540504975212372415, 2.65893092684107510662709321005, 3.25827034717114569518100019075, 3.54110406293956700135370985156, 3.99537293944878661435748904089, 4.50299375581603978348995533716, 4.82604220680341334907637978048, 5.13932141230260486730540470487, 6.00293497616628195120104017780, 6.38428722623152595777496316589, 6.56364957581995654575158709468, 6.75553309758946923431375030008, 7.53204207622829071181453656909, 7.80974681604262524886903360551, 8.396879899200597943631683923191, 8.561939574147500571790315438538, 8.942923879399788595071668107048, 9.185075631281304424842418069504

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