Properties

Label 4-798848-1.1-c1e2-0-0
Degree $4$
Conductor $798848$
Sign $1$
Analytic cond. $50.9352$
Root an. cond. $2.67149$
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-s + 2·3-s + 4-s − 2·6-s − 8-s − 3·9-s + 2·12-s + 16-s + 3·18-s + 4·19-s − 2·24-s − 25-s − 14·27-s − 32-s − 3·36-s − 4·38-s − 24·41-s + 16·43-s + 2·48-s − 13·49-s + 50-s + 14·54-s + 8·57-s − 18·59-s + 64-s − 8·67-s + 3·72-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s − 9-s + 0.577·12-s + 1/4·16-s + 0.707·18-s + 0.917·19-s − 0.408·24-s − 1/5·25-s − 2.69·27-s − 0.176·32-s − 1/2·36-s − 0.648·38-s − 3.74·41-s + 2.43·43-s + 0.288·48-s − 1.85·49-s + 0.141·50-s + 1.90·54-s + 1.05·57-s − 2.34·59-s + 1/8·64-s − 0.977·67-s + 0.353·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798848 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798848 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(798848\)    =    \(2^{7} \cdot 79^{2}\)
Sign: $1$
Analytic conductor: \(50.9352\)
Root analytic conductor: \(2.67149\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 798848,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.452304801\)
\(L(\frac12)\) \(\approx\) \(1.452304801\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( 1 + T \)
79$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
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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

−8.083630712180759633706240971225, −8.008923035003699396981349994067, −7.63212533820846826900119928753, −7.13501126218469989101311707707, −6.43698619599801084891178808154, −6.11894230966477053103157726414, −5.66674862285945396423871921055, −4.96602444926117376624174078421, −4.68407228651939958115867257852, −3.50543676040170862896935659510, −3.35403695021279589879198759951, −3.05075526863841122894839780512, −2.07130866665970371265555871793, −1.90147284566448759002651582081, −0.57937075713009226440087094657, 0.57937075713009226440087094657, 1.90147284566448759002651582081, 2.07130866665970371265555871793, 3.05075526863841122894839780512, 3.35403695021279589879198759951, 3.50543676040170862896935659510, 4.68407228651939958115867257852, 4.96602444926117376624174078421, 5.66674862285945396423871921055, 6.11894230966477053103157726414, 6.43698619599801084891178808154, 7.13501126218469989101311707707, 7.63212533820846826900119928753, 8.008923035003699396981349994067, 8.083630712180759633706240971225

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