↑

Properties

Label 4-623808-1.1-c1e2-0-49
Degree $4$
Conductor $623808$
Sign $-1$
Analytic cond. $39.7745$
Root an. cond. $2.51131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 2·19-s − 2·25-s − 27-s + 6·29-s − 10·41-s + 12·43-s − 10·49-s − 6·53-s + 2·57-s − 16·61-s + 4·71-s + 24·73-s + 2·75-s + 81-s − 6·87-s − 22·89-s − 20·107-s − 10·113-s − 10·121-s + 10·123-s + 127-s − 12·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.458·19-s − 2/5·25-s − 0.192·27-s + 1.11·29-s − 1.56·41-s + 1.82·43-s − 1.42·49-s − 0.824·53-s + 0.264·57-s − 2.04·61-s + 0.474·71-s + 2.80·73-s + 0.230·75-s + 1/9·81-s − 0.643·87-s − 2.33·89-s − 1.93·107-s − 0.940·113-s − 0.909·121-s + 0.901·123-s + 0.0887·127-s − 1.05·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(623808\)    =    \(2^{6} \cdot 3^{3} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(39.7745\)
Root analytic conductor: \(2.51131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
19$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.ag_co
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.k_de
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.43.am_eo
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.g_ek
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.q_fu
67$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \) 2.67.a_aew
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.ae_bu
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.73.ay_la
79$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.79.a_aba
83$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.83.a_abe
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.w_ko
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.a_by
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.170301730162111767394991741203, −7.82032383545000327690015613997, −7.15729612697682940095066724751, −6.69875655142312287785482025094, −6.42846248743658562575909424651, −5.89378190949849651836727246284, −5.40669619693389431610228130992, −4.88009757532814486327692893267, −4.46633381198835408588380617503, −3.93014398078295231043844789834, −3.27603123885212005613003445690, −2.68044833935691010226702237681, −1.90838917985364219921355146561, −1.16627510535619940027438018523, 0, 1.16627510535619940027438018523, 1.90838917985364219921355146561, 2.68044833935691010226702237681, 3.27603123885212005613003445690, 3.93014398078295231043844789834, 4.46633381198835408588380617503, 4.88009757532814486327692893267, 5.40669619693389431610228130992, 5.89378190949849651836727246284, 6.42846248743658562575909424651, 6.69875655142312287785482025094, 7.15729612697682940095066724751, 7.82032383545000327690015613997, 8.170301730162111767394991741203

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