Properties

Label 4-199712-1.1-c1e2-0-1
Degree $4$
Conductor $199712$
Sign $-1$
Analytic cond. $12.7338$
Root an. cond. $1.88903$
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-s + 4-s − 6·5-s + 8-s + 3·9-s − 6·10-s − 10·13-s + 16-s + 12·17-s + 3·18-s − 6·20-s + 17·25-s − 10·26-s + 12·29-s + 32-s + 12·34-s + 3·36-s − 20·37-s − 6·40-s + 4·41-s − 18·45-s − 5·49-s + 17·50-s − 10·52-s − 24·53-s + 12·58-s + 24·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 2.68·5-s + 0.353·8-s + 9-s − 1.89·10-s − 2.77·13-s + 1/4·16-s + 2.91·17-s + 0.707·18-s − 1.34·20-s + 17/5·25-s − 1.96·26-s + 2.22·29-s + 0.176·32-s + 2.05·34-s + 1/2·36-s − 3.28·37-s − 0.948·40-s + 0.624·41-s − 2.68·45-s − 5/7·49-s + 2.40·50-s − 1.38·52-s − 3.29·53-s + 1.57·58-s + 3.07·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 199712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(199712\)    =    \(2^{5} \cdot 79^{2}\)
Sign: $-1$
Analytic conductor: \(12.7338\)
Root analytic conductor: \(1.88903\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{199712} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 199712,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_1$ \( 1 - T \)
79$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
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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

−8.367530614402870047520128581958, −8.316300601517690619428978935480, −7.66138306465432851890178277032, −7.50874638191110861518479439117, −6.92116149525582118764831835825, −6.91576129806276774301284636347, −5.69924671275910195432130153357, −5.03332639383735802564822482656, −4.76932431908835403735443234199, −4.35064038074981651199670304457, −3.45774961440432575864801056504, −3.43543530646035424633948117891, −2.63887435377374729657972230954, −1.31619036555226546624498396933, 0, 1.31619036555226546624498396933, 2.63887435377374729657972230954, 3.43543530646035424633948117891, 3.45774961440432575864801056504, 4.35064038074981651199670304457, 4.76932431908835403735443234199, 5.03332639383735802564822482656, 5.69924671275910195432130153357, 6.91576129806276774301284636347, 6.92116149525582118764831835825, 7.50874638191110861518479439117, 7.66138306465432851890178277032, 8.316300601517690619428978935480, 8.367530614402870047520128581958

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