Properties

Label 4-1183e2-1.1-c1e2-0-5
Degree $4$
Conductor $1399489$
Sign $1$
Analytic cond. $89.2326$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3·3-s + 4·4-s − 3·5-s + 9·6-s − 2·7-s − 3·8-s + 2·9-s + 9·10-s + 3·11-s − 12·12-s + 6·14-s + 9·15-s + 3·16-s + 6·17-s − 6·18-s − 3·19-s − 12·20-s + 6·21-s − 9·22-s + 9·24-s − 2·25-s + 6·27-s − 8·28-s + 3·29-s − 27·30-s − 4·31-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.73·3-s + 2·4-s − 1.34·5-s + 3.67·6-s − 0.755·7-s − 1.06·8-s + 2/3·9-s + 2.84·10-s + 0.904·11-s − 3.46·12-s + 1.60·14-s + 2.32·15-s + 3/4·16-s + 1.45·17-s − 1.41·18-s − 0.688·19-s − 2.68·20-s + 1.30·21-s − 1.91·22-s + 1.83·24-s − 2/5·25-s + 1.15·27-s − 1.51·28-s + 0.557·29-s − 4.92·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1399489\)    =    \(7^{2} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(89.2326\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1183} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1399489,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad7$C_1$ \( ( 1 + T )^{2} \)
13 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T - 9 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 89 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 113 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 12 T + 125 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 121 T^{2} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 21 T + 257 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 31 T + 423 T^{2} + 31 p T^{3} + p^{2} T^{4} \)
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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

−9.513819580291334317990998484185, −9.242418006721246493072843423992, −8.725758535407408592816681592195, −8.404871596259288604045417825207, −7.87717089995583770443176939577, −7.84298405200365218042406102391, −6.97939045327443668455989999066, −6.88671388003764420764706554300, −6.39601888460097042814218355910, −5.85852497429691635704084424954, −5.40272913042532002594487859668, −5.19769500022047440090298103222, −4.11636170255038719593876476212, −3.92723195464011465965801762874, −3.35764240683341862665267275000, −2.62223351766023596639712071955, −1.48942761865667432654331216995, −0.925409949379851576213774966540, 0, 0, 0.925409949379851576213774966540, 1.48942761865667432654331216995, 2.62223351766023596639712071955, 3.35764240683341862665267275000, 3.92723195464011465965801762874, 4.11636170255038719593876476212, 5.19769500022047440090298103222, 5.40272913042532002594487859668, 5.85852497429691635704084424954, 6.39601888460097042814218355910, 6.88671388003764420764706554300, 6.97939045327443668455989999066, 7.84298405200365218042406102391, 7.87717089995583770443176939577, 8.404871596259288604045417825207, 8.725758535407408592816681592195, 9.242418006721246493072843423992, 9.513819580291334317990998484185

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