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Properties

Label 4-777e2-1.1-c1e2-0-23
Degree $4$
Conductor $603729$
Sign $1$
Analytic cond. $38.4942$
Root an. cond. $2.49085$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·7-s − 2·9-s − 4·16-s + 3·21-s + 6·25-s − 5·27-s + 2·37-s + 6·41-s − 6·47-s − 4·48-s + 2·49-s − 6·63-s + 24·67-s + 6·75-s + 81-s + 18·83-s − 6·101-s + 2·111-s − 12·112-s − 13·121-s + 6·123-s + 127-s + 131-s + 137-s + 139-s − 6·141-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s − 2/3·9-s − 16-s + 0.654·21-s + 6/5·25-s − 0.962·27-s + 0.328·37-s + 0.937·41-s − 0.875·47-s − 0.577·48-s + 2/7·49-s − 0.755·63-s + 2.93·67-s + 0.692·75-s + 1/9·81-s + 1.97·83-s − 0.597·101-s + 0.189·111-s − 1.13·112-s − 1.18·121-s + 0.541·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.505·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(603729\)    =    \(3^{2} \cdot 7^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(38.4942\)
Root analytic conductor: \(2.49085\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 603729,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.437519000\)
\(L(\frac12)\) \(\approx\) \(2.437519000\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
7$C_2$ \( 1 - 3 T + p T^{2} \)
37$C_2$ \( 1 - 2 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.a_bq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.41.ag_dn
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.a_fd
73$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.73.a_cn
79$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \) 2.79.a_aes
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.83.as_jn
89$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.89.a_s
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.a_by
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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

−8.321065783179474939822435933353, −8.034529495569077493206218300476, −7.72950229904517849934702995434, −7.04683974805323497879667125600, −6.65333760870998568997782491095, −6.22951641259898399979281636675, −5.43480082495478886410801239456, −5.21206379836143543298628972960, −4.62473571555122321221696594487, −4.18965212376000894080896766771, −3.51615935958112742806595307097, −2.91126118061878046639106846758, −2.31756176087787042795960850559, −1.83051329436985908685428469916, −0.77320653319505038546518008089, 0.77320653319505038546518008089, 1.83051329436985908685428469916, 2.31756176087787042795960850559, 2.91126118061878046639106846758, 3.51615935958112742806595307097, 4.18965212376000894080896766771, 4.62473571555122321221696594487, 5.21206379836143543298628972960, 5.43480082495478886410801239456, 6.22951641259898399979281636675, 6.65333760870998568997782491095, 7.04683974805323497879667125600, 7.72950229904517849934702995434, 8.034529495569077493206218300476, 8.321065783179474939822435933353

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