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Properties

Label 4-819e2-1.1-c1e2-0-25
Degree $4$
Conductor $670761$
Sign $1$
Analytic cond. $42.7683$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 5·7-s + 2·13-s − 2·19-s + 5·25-s + 10·28-s + 7·31-s − 11·37-s − 8·43-s + 18·49-s + 4·52-s − 2·61-s − 8·64-s + 22·67-s + 10·73-s − 4·76-s + 13·79-s + 10·91-s + 19·97-s + 10·100-s − 20·103-s − 2·109-s − 22·121-s + 14·124-s + 127-s + 131-s − 10·133-s + ⋯
L(s)  = 1  + 4-s + 1.88·7-s + 0.554·13-s − 0.458·19-s + 25-s + 1.88·28-s + 1.25·31-s − 1.80·37-s − 1.21·43-s + 18/7·49-s + 0.554·52-s − 0.256·61-s − 64-s + 2.68·67-s + 1.17·73-s − 0.458·76-s + 1.46·79-s + 1.04·91-s + 1.92·97-s + 100-s − 1.97·103-s − 0.191·109-s − 2·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s − 0.867·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(670761\)    =    \(3^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(42.7683\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 670761,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.541184073\)
\(L(\frac12)\) \(\approx\) \(3.541184073\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.l_dg
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.i_v
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.53.a_acb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.67.aw_jv
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.71.a_act
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.ak_bb
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.an_dm
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.97.at_ke
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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

−10.57352848170612887407462873005, −10.47923073334173838096031379118, −9.440413035806214046869547050737, −9.254024404619447129296099950057, −8.409128992400716059858389126806, −8.364181280092702393203494773052, −8.074365763602242258969267276181, −7.42714809638278448126725348425, −6.92730203966564497103370632849, −6.64179616542362120451134178865, −6.24326164681637105960395122523, −5.49077249720267834938449949783, −4.98020458966316439738664840480, −4.87540221413339697965386429516, −4.06380511826387917448556425493, −3.57826824341302565856737080272, −2.78439797316068923609797133448, −2.17780326022769010648367878704, −1.72887169757632256400104817440, −1.00098490011391883988961392684, 1.00098490011391883988961392684, 1.72887169757632256400104817440, 2.17780326022769010648367878704, 2.78439797316068923609797133448, 3.57826824341302565856737080272, 4.06380511826387917448556425493, 4.87540221413339697965386429516, 4.98020458966316439738664840480, 5.49077249720267834938449949783, 6.24326164681637105960395122523, 6.64179616542362120451134178865, 6.92730203966564497103370632849, 7.42714809638278448126725348425, 8.074365763602242258969267276181, 8.364181280092702393203494773052, 8.409128992400716059858389126806, 9.254024404619447129296099950057, 9.440413035806214046869547050737, 10.47923073334173838096031379118, 10.57352848170612887407462873005

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