↑

Properties

Label 4-623808-1.1-c1e2-0-2
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s − 4·13-s + 4·21-s + 2·25-s − 27-s + 12·31-s + 4·39-s + 2·49-s − 20·61-s − 4·63-s − 4·67-s − 12·73-s − 2·75-s − 8·79-s + 81-s + 16·91-s − 12·93-s + 28·97-s − 28·109-s − 4·117-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.10·13-s + 0.872·21-s + 2/5·25-s − 0.192·27-s + 2.15·31-s + 0.640·39-s + 2/7·49-s − 2.56·61-s − 0.503·63-s − 0.488·67-s − 1.40·73-s − 0.230·75-s − 0.900·79-s + 1/9·81-s + 1.67·91-s − 1.24·93-s + 2.84·97-s − 2.68·109-s − 0.369·117-s + 1.27·121-s + 0.0887·127-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: \(0\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7301219763\)
\(L(\frac12)\) \(\approx\) \(0.7301219763\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.e_o
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.am_dq
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.41.a_acw
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.47.a_bi
53$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.53.a_w
59$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.59.a_acg
61$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.u_hy
67$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.e_fe
71$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.71.a_ade
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.m_eo
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.i_be
83$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.83.a_bi
89$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.89.a_acg
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.97.abc_ok
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

−8.332827498282832737739712254040, −7.927678245907970024983186163171, −7.22520814590044657103192726987, −7.13745162637463786490563907108, −6.44241646790166109966282863223, −6.18312063456974917729884758292, −5.82873054307586127933097371351, −5.08355466714749303651769512594, −4.60554725500317971871382121397, −4.31999733292339050476475376099, −3.40149734621719752645486836365, −2.99239433136804527003910004770, −2.52654563932471743750083244413, −1.51703106423776465772975343983, −0.44858401288340263293785916832, 0.44858401288340263293785916832, 1.51703106423776465772975343983, 2.52654563932471743750083244413, 2.99239433136804527003910004770, 3.40149734621719752645486836365, 4.31999733292339050476475376099, 4.60554725500317971871382121397, 5.08355466714749303651769512594, 5.82873054307586127933097371351, 6.18312063456974917729884758292, 6.44241646790166109966282863223, 7.13745162637463786490563907108, 7.22520814590044657103192726987, 7.927678245907970024983186163171, 8.332827498282832737739712254040

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