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Properties

Label 4-450e2-1.1-c1e2-0-7
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $12.9115$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 3·6-s + 7-s + 8-s + 6·9-s + 2·11-s + 6·13-s − 14-s − 16-s − 4·17-s − 6·18-s + 12·19-s − 3·21-s − 2·22-s − 23-s − 3·24-s − 6·26-s − 9·27-s − 9·29-s + 2·31-s − 6·33-s + 4·34-s + 4·37-s − 12·38-s − 18·39-s + 11·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.603·11-s + 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s − 1.41·18-s + 2.75·19-s − 0.654·21-s − 0.426·22-s − 0.208·23-s − 0.612·24-s − 1.17·26-s − 1.73·27-s − 1.67·29-s + 0.359·31-s − 1.04·33-s + 0.685·34-s + 0.657·37-s − 1.94·38-s − 2.88·39-s + 1.71·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(12.9115\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8080909771\)
\(L(\frac12)\) \(\approx\) \(0.8080909771\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) 2.23.b_aw
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_abb
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 11 T + 80 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.41.al_dc
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_abb
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.47.ah_c
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_abr
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.61.ah_am
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.al_cc
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_cn
83$C_2^2$ \( 1 - 11 T + 38 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.83.al_bm
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.89.ac_gx
97$C_2^2$ \( 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.97.ai_abh
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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

−11.25219571113539281945455149920, −11.04726805948590608931520981806, −10.42720379826972787954285094758, −10.16431134189938566569845765730, −9.334755061906052980727749152159, −9.273141675490761525203275316781, −8.827790767311597196077133577803, −8.033183508616444963954119517298, −7.40462924476539799843451072119, −7.40394262571530211538012265832, −6.59864655247717234300780869072, −6.06685545682879593289550242577, −5.75115432739394494882569238313, −5.26664752326488959808064646856, −4.68192824884151139894029836659, −3.92029443215693296252491765638, −3.69376433286278946803766917838, −2.36285156628891161669948659412, −1.22545671346818461640141781313, −0.894228300865341295673256873882, 0.894228300865341295673256873882, 1.22545671346818461640141781313, 2.36285156628891161669948659412, 3.69376433286278946803766917838, 3.92029443215693296252491765638, 4.68192824884151139894029836659, 5.26664752326488959808064646856, 5.75115432739394494882569238313, 6.06685545682879593289550242577, 6.59864655247717234300780869072, 7.40394262571530211538012265832, 7.40462924476539799843451072119, 8.033183508616444963954119517298, 8.827790767311597196077133577803, 9.273141675490761525203275316781, 9.334755061906052980727749152159, 10.16431134189938566569845765730, 10.42720379826972787954285094758, 11.04726805948590608931520981806, 11.25219571113539281945455149920

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