Properties

Label 4-555579-1.1-c1e2-0-9
Degree $4$
Conductor $555579$
Sign $-1$
Analytic cond. $35.4241$
Root an. cond. $2.43963$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·7-s − 4·8-s + 4·14-s − 4·16-s − 19-s − 9·25-s − 2·29-s − 2·38-s + 4·41-s + 2·43-s − 7·49-s − 18·50-s − 10·53-s − 8·56-s − 4·58-s + 12·59-s − 2·61-s + 14·73-s + 8·82-s + 4·86-s − 10·89-s − 14·98-s − 20·106-s − 26·107-s − 8·112-s − 22·113-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.755·7-s − 1.41·8-s + 1.06·14-s − 16-s − 0.229·19-s − 9/5·25-s − 0.371·29-s − 0.324·38-s + 0.624·41-s + 0.304·43-s − 49-s − 2.54·50-s − 1.37·53-s − 1.06·56-s − 0.525·58-s + 1.56·59-s − 0.256·61-s + 1.63·73-s + 0.883·82-s + 0.431·86-s − 1.05·89-s − 1.41·98-s − 1.94·106-s − 2.51·107-s − 0.755·112-s − 2.06·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(555579\)    =    \(3^{4} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(35.4241\)
Root analytic conductor: \(2.43963\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 555579,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3 \( 1 \)
19$C_1$ \( 1 + T \)
good2$C_2$$\times$$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 81 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
79$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
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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

−8.160281867836798413988504593472, −7.82231118705235532431700121579, −7.35927819633061108106985748636, −6.61686854924244755386600433480, −6.24534433215891451039183862188, −5.71236417426004027386072680006, −5.24824148924913907463470807081, −5.01040280724707070956568436462, −4.36261738315499567417804023485, −3.95166374584408301169162577766, −3.68104841789891065387354576043, −2.87870936617689303653927836138, −2.20576342724070175583602554471, −1.40956406516392261471269748811, 0, 1.40956406516392261471269748811, 2.20576342724070175583602554471, 2.87870936617689303653927836138, 3.68104841789891065387354576043, 3.95166374584408301169162577766, 4.36261738315499567417804023485, 5.01040280724707070956568436462, 5.24824148924913907463470807081, 5.71236417426004027386072680006, 6.24534433215891451039183862188, 6.61686854924244755386600433480, 7.35927819633061108106985748636, 7.82231118705235532431700121579, 8.160281867836798413988504593472

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