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Properties

Label 4-882e2-1.1-c1e2-0-23
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

\(L(1)\) \(\approx\) \(3.089441585\)
\(L(\frac12)\) \(\approx\) \(3.089441585\)
\(L(\frac{3}{2})\) 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.b_ae
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_ah
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ah_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.ai_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.am_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.q_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.ai_fe
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.61.aba_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.ao_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.m_cj
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_ed
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

−10.25396209579769943462072331811, −10.24842505073903847604436003636, −9.787730416145806689417618209518, −9.326112427445416328035364320842, −8.495074317128414875875664217310, −8.051663524382363805446190205549, −7.53366370039563382079210199850, −7.13999089909872014995369911906, −6.58060744470450375492320112795, −6.51498100374461079882812658438, −5.87874626957330715393465897126, −5.50351430442072685595884490527, −4.98579465736579177885288479322, −4.73613154969714712791934883050, −4.02514432846578855863203501648, −4.00724754078071634669650158367, −2.77221069967928847209810332960, −2.74718937281670050114318812260, −1.34448496955604964700859277713, −0.865013661055320781129007301315, 0.865013661055320781129007301315, 1.34448496955604964700859277713, 2.74718937281670050114318812260, 2.77221069967928847209810332960, 4.00724754078071634669650158367, 4.02514432846578855863203501648, 4.73613154969714712791934883050, 4.98579465736579177885288479322, 5.50351430442072685595884490527, 5.87874626957330715393465897126, 6.51498100374461079882812658438, 6.58060744470450375492320112795, 7.13999089909872014995369911906, 7.53366370039563382079210199850, 8.051663524382363805446190205549, 8.495074317128414875875664217310, 9.326112427445416328035364320842, 9.787730416145806689417618209518, 10.24842505073903847604436003636, 10.25396209579769943462072331811

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