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Properties

Label 4-1850e2-1.1-c1e2-0-24
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s + 3·4-s + 8·6-s + 3·7-s − 4·8-s + 6·9-s − 11-s − 12·12-s − 2·13-s − 6·14-s + 5·16-s + 5·17-s − 12·18-s − 4·19-s − 12·21-s + 2·22-s − 2·23-s + 16·24-s + 4·26-s + 4·27-s + 9·28-s − 29-s − 11·31-s − 6·32-s + 4·33-s − 10·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 3/2·4-s + 3.26·6-s + 1.13·7-s − 1.41·8-s + 2·9-s − 0.301·11-s − 3.46·12-s − 0.554·13-s − 1.60·14-s + 5/4·16-s + 1.21·17-s − 2.82·18-s − 0.917·19-s − 2.61·21-s + 0.426·22-s − 0.417·23-s + 3.26·24-s + 0.784·26-s + 0.769·27-s + 1.70·28-s − 0.185·29-s − 1.97·31-s − 1.06·32-s + 0.696·33-s − 1.71·34-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: \(2\)
Selberg data: \((4,\ 3422500,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
37$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
7$D_{4}$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_i
11$D_{4}$ \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_o
13$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_ag
17$D_{4}$ \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_bg
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_o
29$D_{4}$ \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_aq
31$D_{4}$ \( 1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.31.l_dg
41$D_{4}$ \( 1 - 5 T + 80 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.41.af_dc
43$D_{4}$ \( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.43.j_du
47$D_{4}$ \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_cs
53$D_{4}$ \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_dw
59$D_{4}$ \( 1 - 14 T + 134 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_fe
61$D_{4}$ \( 1 + 5 T + 120 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_eq
67$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_dy
71$D_{4}$ \( 1 + 2 T + 110 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.71.c_eg
73$D_{4}$ \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_es
79$D_{4}$ \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_ew
83$D_{4}$ \( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_gc
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.97.r_hk
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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

−8.940320223711762962223230912755, −8.737052205505259617471811452516, −8.156267292721367562811560124814, −7.985374050494055711862611586640, −7.28981585347324473914742674662, −7.28049098393576106993935962498, −6.66030687927452652207090414995, −6.19036550881638327218872196326, −5.97709781165528597738580603587, −5.31705363608102308496089227500, −5.19638799615103081115813939715, −5.02561438281820363204533597119, −4.04333333034968813031469823554, −3.76187398807366899802760505357, −2.69766668476017202650903454300, −2.37466375300647471221377237867, −1.42281832082459148997276125158, −1.21018847376649232357277933766, 0, 0, 1.21018847376649232357277933766, 1.42281832082459148997276125158, 2.37466375300647471221377237867, 2.69766668476017202650903454300, 3.76187398807366899802760505357, 4.04333333034968813031469823554, 5.02561438281820363204533597119, 5.19638799615103081115813939715, 5.31705363608102308496089227500, 5.97709781165528597738580603587, 6.19036550881638327218872196326, 6.66030687927452652207090414995, 7.28049098393576106993935962498, 7.28981585347324473914742674662, 7.985374050494055711862611586640, 8.156267292721367562811560124814, 8.737052205505259617471811452516, 8.940320223711762962223230912755

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