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Properties

Label 4-882e2-1.1-c1e2-0-42
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 6·11-s + 5·16-s − 12·22-s + 12·23-s − 25-s − 18·29-s − 6·32-s − 20·37-s − 8·43-s + 18·44-s − 24·46-s + 2·50-s + 6·53-s + 36·58-s + 7·64-s + 4·67-s + 40·74-s + 10·79-s + 16·86-s − 24·88-s + 36·92-s − 3·100-s − 12·106-s − 6·107-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1.80·11-s + 5/4·16-s − 2.55·22-s + 2.50·23-s − 1/5·25-s − 3.34·29-s − 1.06·32-s − 3.28·37-s − 1.21·43-s + 2.71·44-s − 3.53·46-s + 0.282·50-s + 0.824·53-s + 4.72·58-s + 7/8·64-s + 0.488·67-s + 4.64·74-s + 1.12·79-s + 1.72·86-s − 2.55·88-s + 3.75·92-s − 0.299·100-s − 1.16·106-s − 0.580·107-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: \(1\)
Selberg data: \((4,\ 777924,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.11.ag_bf
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.29.s_fj
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.a_n
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.a_ef
61$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.61.a_ec
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.79.ak_hb
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.83.a_dh
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.97.a_z
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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

−8.143428262544059580201771154193, −7.56006473562539600008273040402, −7.14400221617286508047479839089, −6.76340153457321918972152567863, −6.70720744145358122288448799487, −5.88431208298112457486858305913, −5.30135083620848369798403743062, −5.10036072121782338740270472262, −4.05162352634345833644814800792, −3.52994453436524682105467711302, −3.32944386865340105186531370454, −2.28166780967037125049719495570, −1.64799092613985375020716144538, −1.22415852747718716380535497945, 0, 1.22415852747718716380535497945, 1.64799092613985375020716144538, 2.28166780967037125049719495570, 3.32944386865340105186531370454, 3.52994453436524682105467711302, 4.05162352634345833644814800792, 5.10036072121782338740270472262, 5.30135083620848369798403743062, 5.88431208298112457486858305913, 6.70720744145358122288448799487, 6.76340153457321918972152567863, 7.14400221617286508047479839089, 7.56006473562539600008273040402, 8.143428262544059580201771154193

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