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Properties

Label 4-1080000-1.1-c1e2-0-12
Degree $4$
Conductor $1080000$
Sign $-1$
Analytic cond. $68.8617$
Root an. cond. $2.88067$
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  + 3-s − 4·7-s + 9-s + 4·13-s + 16·19-s − 4·21-s + 27-s − 20·37-s + 4·39-s − 24·43-s − 2·49-s + 16·57-s + 4·61-s − 4·63-s − 16·67-s + 8·73-s − 16·79-s + 81-s − 16·91-s + 16·97-s − 4·103-s − 12·109-s − 20·111-s + 4·117-s − 18·121-s + 127-s − 24·129-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.10·13-s + 3.67·19-s − 0.872·21-s + 0.192·27-s − 3.28·37-s + 0.640·39-s − 3.65·43-s − 2/7·49-s + 2.11·57-s + 0.512·61-s − 0.503·63-s − 1.95·67-s + 0.936·73-s − 1.80·79-s + 1/9·81-s − 1.67·91-s + 1.62·97-s − 0.394·103-s − 1.14·109-s − 1.89·111-s + 0.369·117-s − 1.63·121-s + 0.0887·127-s − 2.11·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1080000\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(68.8617\)
Root analytic conductor: \(2.88067\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1080000,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 - T \)
5 \( 1 \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.19.aq_dy
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.a_da
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.a_g
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$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.67.q_hq
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−8.065360238159720695767955399926, −7.15562382903539764269849136934, −7.15109339700237674664040328399, −6.63560342158760211146375054979, −6.20989492713295015084463586012, −5.56718290716607818845063804153, −5.14891659899478257407149456127, −4.90786055385638384453526942829, −3.68478607534762393615004752252, −3.63702338059271712194862462334, −3.07956734277475342949311345221, −2.98769726639906032012813910091, −1.65944442938371927957526732518, −1.32041558029711473446320898964, 0, 1.32041558029711473446320898964, 1.65944442938371927957526732518, 2.98769726639906032012813910091, 3.07956734277475342949311345221, 3.63702338059271712194862462334, 3.68478607534762393615004752252, 4.90786055385638384453526942829, 5.14891659899478257407149456127, 5.56718290716607818845063804153, 6.20989492713295015084463586012, 6.63560342158760211146375054979, 7.15109339700237674664040328399, 7.15562382903539764269849136934, 8.065360238159720695767955399926

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