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Properties

Label 4-243675-1.1-c1e2-0-1
Degree $4$
Conductor $243675$
Sign $1$
Analytic cond. $15.5369$
Root an. cond. $1.98536$
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 − 3·4-s + 8·7-s + 9-s + 3·12-s + 4·13-s + 5·16-s − 2·19-s − 8·21-s + 25-s − 27-s − 24·28-s − 3·36-s − 12·37-s − 4·39-s + 16·43-s − 5·48-s + 34·49-s − 12·52-s + 2·57-s + 28·61-s + 8·63-s − 3·64-s − 8·67-s − 28·73-s − 75-s + 6·76-s + ⋯
L(s)  = 1  − 0.577·3-s − 3/2·4-s + 3.02·7-s + 1/3·9-s + 0.866·12-s + 1.10·13-s + 5/4·16-s − 0.458·19-s − 1.74·21-s + 1/5·25-s − 0.192·27-s − 4.53·28-s − 1/2·36-s − 1.97·37-s − 0.640·39-s + 2.43·43-s − 0.721·48-s + 34/7·49-s − 1.66·52-s + 0.264·57-s + 3.58·61-s + 1.00·63-s − 3/8·64-s − 0.977·67-s − 3.27·73-s − 0.115·75-s + 0.688·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(243675\)    =    \(3^{3} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(15.5369\)
Root analytic conductor: \(1.98536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 243675,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.533147475\)
\(L(\frac12)\) \(\approx\) \(1.533147475\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.a_adm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.61.abc_mg
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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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

−8.956131697052494305782515642843, −8.576861436588155713465515075006, −8.056447043933612031171269197880, −7.79111029211693930669614606595, −7.22090991652715915706448156971, −6.54655367252788500605108631348, −5.59885321084019164088416378698, −5.57442263951836682011583202205, −4.80915338219819536204131275847, −4.73512523477553586192350141458, −4.00775128266596845974693279239, −3.75150945152824378664368493386, −2.32830304228468473527730093464, −1.55903104009719190466896724014, −0.908717833826787650120905118138, 0.908717833826787650120905118138, 1.55903104009719190466896724014, 2.32830304228468473527730093464, 3.75150945152824378664368493386, 4.00775128266596845974693279239, 4.73512523477553586192350141458, 4.80915338219819536204131275847, 5.57442263951836682011583202205, 5.59885321084019164088416378698, 6.54655367252788500605108631348, 7.22090991652715915706448156971, 7.79111029211693930669614606595, 8.056447043933612031171269197880, 8.576861436588155713465515075006, 8.956131697052494305782515642843

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