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Properties

Label 4-882e2-1.1-c1e2-0-10
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $49.6011$
Root an. cond. $2.65382$
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-s − 3·5-s − 8-s − 3·9-s − 3·10-s + 6·11-s + 2·13-s − 16-s − 12·17-s − 3·18-s + 14·19-s + 6·22-s − 3·23-s + 5·25-s + 2·26-s − 6·29-s + 2·31-s − 12·34-s + 4·37-s + 14·38-s + 3·40-s − 2·43-s + 9·45-s − 3·46-s + 5·50-s + 12·53-s − 18·55-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.34·5-s − 0.353·8-s − 9-s − 0.948·10-s + 1.80·11-s + 0.554·13-s − 1/4·16-s − 2.91·17-s − 0.707·18-s + 3.21·19-s + 1.27·22-s − 0.625·23-s + 25-s + 0.392·26-s − 1.11·29-s + 0.359·31-s − 2.05·34-s + 0.657·37-s + 2.27·38-s + 0.474·40-s − 0.304·43-s + 1.34·45-s − 0.442·46-s + 0.707·50-s + 1.64·53-s − 2.42·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(49.6011\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.683159637\)
\(L(\frac12)\) \(\approx\) \(1.683159637\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_e
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_z
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.ac_aj
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.19.ao_dj
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ao
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_h
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_abb
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_abn
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.61.af_abk
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_ad
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.71.ag_fv
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.79.f_acc
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−10.59694363920667051924635489392, −9.736000199304687786589764240958, −9.290633417278735376827883738495, −9.117445548242253407093474649899, −8.789726955683827053751237734212, −8.175858267806827663603888974826, −7.899321443447152261510493070964, −7.19623098128942279082882958680, −6.96945011160415676378517493244, −6.42009486056554632692770233050, −6.08502832549559124301236003416, −5.33142328398286281969769102218, −5.14872375829298971321837624691, −4.24839739764351505823086831606, −4.06336083359052502233310536445, −3.71897462971167064660627416686, −3.09115944200524700898693665764, −2.55538912827991486450575441755, −1.54597252019072772210147034697, −0.58195978553675455293024134787, 0.58195978553675455293024134787, 1.54597252019072772210147034697, 2.55538912827991486450575441755, 3.09115944200524700898693665764, 3.71897462971167064660627416686, 4.06336083359052502233310536445, 4.24839739764351505823086831606, 5.14872375829298971321837624691, 5.33142328398286281969769102218, 6.08502832549559124301236003416, 6.42009486056554632692770233050, 6.96945011160415676378517493244, 7.19623098128942279082882958680, 7.899321443447152261510493070964, 8.175858267806827663603888974826, 8.789726955683827053751237734212, 9.117445548242253407093474649899, 9.290633417278735376827883738495, 9.736000199304687786589764240958, 10.59694363920667051924635489392

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