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Properties

Label 4-777e2-1.1-c1e2-0-3
Degree $4$
Conductor $603729$
Sign $1$
Analytic cond. $38.4942$
Root an. cond. $2.49085$
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  + 3·3-s − 2·4-s − 5·7-s + 6·9-s − 6·12-s − 9·13-s − 15·21-s − 10·25-s + 9·27-s + 10·28-s + 3·31-s − 12·36-s + 10·37-s − 27·39-s − 16·43-s + 18·49-s + 18·52-s − 30·63-s + 8·64-s − 5·67-s + 3·73-s − 30·75-s + 34·79-s + 9·81-s + 30·84-s + 45·91-s + 9·93-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s − 1.88·7-s + 2·9-s − 1.73·12-s − 2.49·13-s − 3.27·21-s − 2·25-s + 1.73·27-s + 1.88·28-s + 0.538·31-s − 2·36-s + 1.64·37-s − 4.32·39-s − 2.43·43-s + 18/7·49-s + 2.49·52-s − 3.77·63-s + 64-s − 0.610·67-s + 0.351·73-s − 3.46·75-s + 3.82·79-s + 81-s + 3.27·84-s + 4.71·91-s + 0.933·93-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 603729,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.055990222\)
\(L(\frac12)\) \(\approx\) \(1.055990222\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
37$C_2$ \( 1 - 10 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.j_bo
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ad_bi
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.a_acw
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.71.a_ct
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.ad_cy
79$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.79.abi_rf
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.a_gn
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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

−10.24839410569640360771890861367, −9.704766783408996358295623567601, −9.680575689277084583703114949302, −9.289238742132577883839838902371, −9.133076436879259073891665810764, −8.334252719304444529227826274857, −8.060266070560386196087412026357, −7.53633778876488505355119721477, −7.31912851141035435706283555839, −6.50192044663494780148870094389, −6.48825850855240248749710859238, −5.54042192951823015483651674087, −4.99321505315219389416143148227, −4.44482638401852751643074842468, −4.06935044484267778337421263085, −3.37718276873004773960386884762, −3.15033951784934076351455515258, −2.28752783874445607201004407314, −2.16607096120113883519053262486, −0.44401093016556136145152491538, 0.44401093016556136145152491538, 2.16607096120113883519053262486, 2.28752783874445607201004407314, 3.15033951784934076351455515258, 3.37718276873004773960386884762, 4.06935044484267778337421263085, 4.44482638401852751643074842468, 4.99321505315219389416143148227, 5.54042192951823015483651674087, 6.48825850855240248749710859238, 6.50192044663494780148870094389, 7.31912851141035435706283555839, 7.53633778876488505355119721477, 8.060266070560386196087412026357, 8.334252719304444529227826274857, 9.133076436879259073891665810764, 9.289238742132577883839838902371, 9.680575689277084583703114949302, 9.704766783408996358295623567601, 10.24839410569640360771890861367

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