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Properties

Label 4-882e2-1.1-c1e2-0-52
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 3·4-s + 5-s + 6·6-s + 4·8-s + 6·9-s + 2·10-s + 2·11-s + 9·12-s + 2·13-s + 3·15-s + 5·16-s + 12·18-s − 7·19-s + 3·20-s + 4·22-s − 3·23-s + 12·24-s + 5·25-s + 4·26-s + 9·27-s + 8·29-s + 6·30-s − 8·31-s + 6·32-s + 6·33-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.73·3-s + 3/2·4-s + 0.447·5-s + 2.44·6-s + 1.41·8-s + 2·9-s + 0.632·10-s + 0.603·11-s + 2.59·12-s + 0.554·13-s + 0.774·15-s + 5/4·16-s + 2.82·18-s − 1.60·19-s + 0.670·20-s + 0.852·22-s − 0.625·23-s + 2.44·24-s + 25-s + 0.784·26-s + 1.73·27-s + 1.48·29-s + 1.09·30-s − 1.43·31-s + 1.06·32-s + 1.04·33-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

\(L(1)\) \(\approx\) \(11.50040835\)
\(L(\frac12)\) \(\approx\) \(11.50040835\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3$C_2$ \( 1 - p T + p T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.ac_aj
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.h_be
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ao
29$C_2^2$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_bj
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_ab
41$C_2^2$ \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_dz
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.53.e_abl
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.61.ba_lf
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.67.e_fi
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.71.k_gl
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.73.o_et
79$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.79.w_kt
83$C_2^2$ \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.83.am_cj
89$C_2^2$ \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.89.ao_ed
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_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

−10.24928735491817010990957004318, −10.20146749885659175990741130290, −9.198450010536496057141673274143, −9.080708600348825569685490039945, −8.763741754387745562606186147565, −8.278154589518946319447741292519, −7.57213595954818906173831958355, −7.53520605461868283020667092820, −6.81944051310264722369827652064, −6.41835757670828001597970539845, −6.02099304768654761605733423058, −5.63479974483130136865129532210, −4.60337356946091157326254310396, −4.52277536523174472076842759039, −4.03548871328453566246913551667, −3.51325462622673589719819883675, −2.80455990106201792401222549283, −2.71671479382720217630510662210, −1.77528803446710747546397883838, −1.47567785996464988209168649710, 1.47567785996464988209168649710, 1.77528803446710747546397883838, 2.71671479382720217630510662210, 2.80455990106201792401222549283, 3.51325462622673589719819883675, 4.03548871328453566246913551667, 4.52277536523174472076842759039, 4.60337356946091157326254310396, 5.63479974483130136865129532210, 6.02099304768654761605733423058, 6.41835757670828001597970539845, 6.81944051310264722369827652064, 7.53520605461868283020667092820, 7.57213595954818906173831958355, 8.278154589518946319447741292519, 8.763741754387745562606186147565, 9.080708600348825569685490039945, 9.198450010536496057141673274143, 10.20146749885659175990741130290, 10.24928735491817010990957004318

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