↑

Properties

Label 4-777e2-1.1-c1e2-0-13
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 4·7-s − 3·9-s − 4·11-s + 5·16-s + 2·25-s − 12·28-s − 9·36-s + 6·37-s − 8·41-s − 12·44-s − 16·47-s + 9·49-s + 16·53-s + 12·63-s + 3·64-s − 8·71-s + 28·73-s + 16·77-s + 9·81-s + 8·83-s + 12·99-s + 6·100-s + 8·101-s + 8·107-s − 20·112-s − 6·121-s + ⋯
L(s)  = 1  + 3/2·4-s − 1.51·7-s − 9-s − 1.20·11-s + 5/4·16-s + 2/5·25-s − 2.26·28-s − 3/2·36-s + 0.986·37-s − 1.24·41-s − 1.80·44-s − 2.33·47-s + 9/7·49-s + 2.19·53-s + 1.51·63-s + 3/8·64-s − 0.949·71-s + 3.27·73-s + 1.82·77-s + 81-s + 0.878·83-s + 1.20·99-s + 3/5·100-s + 0.796·101-s + 0.773·107-s − 1.88·112-s − 0.545·121-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

\(L(1)\) \(\approx\) \(1.515453102\)
\(L(\frac12)\) \(\approx\) \(1.515453102\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
37$C_2$ \( 1 - 6 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.e_w
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.23.a_ag
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.31.a_g
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.i_ck
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.aq_gk
59$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.59.a_ag
61$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.61.a_ao
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.i_fm
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.89.a_be
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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

−8.213071737540256960352572016657, −8.041771390690434230737787313983, −7.34600041817173219165717323104, −7.04912388169264662518464809687, −6.43127151070208806055186331771, −6.34818869886641312808957189532, −5.80493175472966495170573698895, −5.24888090740380272281329342041, −4.88461289229920871087832768725, −3.83887086887814662235394780648, −3.31478131886312285618798210207, −2.93428679795690465335967583405, −2.47725689837240047387960069750, −1.92341591010950894892246881327, −0.57418614117802865819084681195, 0.57418614117802865819084681195, 1.92341591010950894892246881327, 2.47725689837240047387960069750, 2.93428679795690465335967583405, 3.31478131886312285618798210207, 3.83887086887814662235394780648, 4.88461289229920871087832768725, 5.24888090740380272281329342041, 5.80493175472966495170573698895, 6.34818869886641312808957189532, 6.43127151070208806055186331771, 7.04912388169264662518464809687, 7.34600041817173219165717323104, 8.041771390690434230737787313983, 8.213071737540256960352572016657

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