Properties

Label 4-64827-1.1-c1e2-0-0
Degree $4$
Conductor $64827$
Sign $1$
Analytic cond. $4.13342$
Root an. cond. $1.42586$
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  − 3-s + 4·5-s + 9-s − 4·15-s − 4·16-s + 2·25-s − 27-s + 6·37-s + 20·41-s + 10·43-s + 4·45-s + 12·47-s + 4·48-s + 24·59-s − 10·67-s − 2·75-s − 2·79-s − 16·80-s + 81-s − 12·83-s − 32·89-s − 4·101-s + 18·109-s − 6·111-s − 18·121-s − 20·123-s − 28·125-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s + 1/3·9-s − 1.03·15-s − 16-s + 2/5·25-s − 0.192·27-s + 0.986·37-s + 3.12·41-s + 1.52·43-s + 0.596·45-s + 1.75·47-s + 0.577·48-s + 3.12·59-s − 1.22·67-s − 0.230·75-s − 0.225·79-s − 1.78·80-s + 1/9·81-s − 1.31·83-s − 3.39·89-s − 0.398·101-s + 1.72·109-s − 0.569·111-s − 1.63·121-s − 1.80·123-s − 2.50·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64827 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(64827\)    =    \(3^{3} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4.13342\)
Root analytic conductor: \(1.42586\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 64827,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.608966934\)
\(L(\frac12)\) \(\approx\) \(1.608966934\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_1$ \( 1 + T \)
7 \( 1 \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.a_s
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.a_bl
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.a_at
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.43.ak_eh
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.a_abm
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.67.k_gd
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.73.a_fh
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.89.bg_qs
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.a_gc
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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

−9.879400023159387748546449740681, −9.459031124333216147077448358277, −9.092563983560312555964020990816, −8.583706845901838358159961836152, −7.67312246287158311998952783143, −7.29485403540511127428179965078, −6.65130832285966559117996014878, −6.01621490934601452163404663718, −5.72037257717675019727399870073, −5.40855511243246313730427714163, −4.23783192793666623809357478808, −4.18739539622885934722496613081, −2.47062022887811000287520417928, −2.43327032489084752763662196914, −1.15383453307155328641933378041, 1.15383453307155328641933378041, 2.43327032489084752763662196914, 2.47062022887811000287520417928, 4.18739539622885934722496613081, 4.23783192793666623809357478808, 5.40855511243246313730427714163, 5.72037257717675019727399870073, 6.01621490934601452163404663718, 6.65130832285966559117996014878, 7.29485403540511127428179965078, 7.67312246287158311998952783143, 8.583706845901838358159961836152, 9.092563983560312555964020990816, 9.459031124333216147077448358277, 9.879400023159387748546449740681

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