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Properties

Label 4-450e2-1.1-c1e2-0-13
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·7-s + 8·13-s + 16-s − 8·19-s − 4·28-s − 8·31-s − 16·37-s − 16·43-s − 2·49-s + 8·52-s + 4·61-s + 64-s + 8·67-s + 20·73-s − 8·76-s − 8·79-s − 32·91-s − 4·97-s − 4·103-s + 4·109-s − 4·112-s + 14·121-s − 8·124-s + 127-s + 131-s + 32·133-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.51·7-s + 2.21·13-s + 1/4·16-s − 1.83·19-s − 0.755·28-s − 1.43·31-s − 2.63·37-s − 2.43·43-s − 2/7·49-s + 1.10·52-s + 0.512·61-s + 1/8·64-s + 0.977·67-s + 2.34·73-s − 0.917·76-s − 0.900·79-s − 3.35·91-s − 0.406·97-s − 0.394·103-s + 0.383·109-s − 0.377·112-s + 1.27·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-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: \(1\)
Selberg data: \((4,\ 202500,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.696574884545133458339458478384, −8.428134066544604724277886211493, −8.168778969487982270877121302424, −7.05855949257969394331275338730, −6.90983690463256235300331271327, −6.35337453909666730560193822981, −6.18542098793921066559307328051, −5.50042078426760473642213736203, −4.91552207061368779924910287591, −3.93114117360591390680316014330, −3.44283204101776149845879223587, −3.38140910088455755067648438945, −2.17323488972164827096525475937, −1.55715684978414451776466943001, 0, 1.55715684978414451776466943001, 2.17323488972164827096525475937, 3.38140910088455755067648438945, 3.44283204101776149845879223587, 3.93114117360591390680316014330, 4.91552207061368779924910287591, 5.50042078426760473642213736203, 6.18542098793921066559307328051, 6.35337453909666730560193822981, 6.90983690463256235300331271327, 7.05855949257969394331275338730, 8.168778969487982270877121302424, 8.428134066544604724277886211493, 8.696574884545133458339458478384

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