Properties

Label 4-882e2-1.1-c1e2-0-48
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 3·5-s + 3·6-s + 8-s + 6·9-s + 3·10-s + 3·11-s + 5·13-s + 9·15-s − 16-s − 6·17-s − 6·18-s − 10·19-s − 3·22-s + 3·23-s − 3·24-s + 5·25-s − 5·26-s − 9·27-s + 3·29-s − 9·30-s − 4·31-s − 9·33-s + 6·34-s − 14·37-s + 10·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 1.34·5-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.948·10-s + 0.904·11-s + 1.38·13-s + 2.32·15-s − 1/4·16-s − 1.45·17-s − 1.41·18-s − 2.29·19-s − 0.639·22-s + 0.625·23-s − 0.612·24-s + 25-s − 0.980·26-s − 1.73·27-s + 0.557·29-s − 1.64·30-s − 0.718·31-s − 1.56·33-s + 1.02·34-s − 2.30·37-s + 1.62·38-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: \(2\)
Selberg data: \((4,\ 777924,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + T + T^{2} \)
3$C_2$ \( 1 + p T + p T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \)
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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

−10.19562869270034713479775873560, −9.523756629809606465296259622115, −8.794416419967053015032500806870, −8.724726512086160746890719913753, −8.406572554738207186652410747970, −7.973657625687070944670092759898, −7.00337385015735356382566383571, −6.88102877667591105738239412144, −6.63130700951211029382758832697, −6.31161855661158165100669513573, −5.40240999221953605980898689631, −5.20092372061562563148816864730, −4.40176178283255397633132059669, −4.08761638957052283837292033521, −3.90816121421174497149503325854, −3.01586743033391632798713887408, −1.73913013012394302611000660936, −1.38710966815627223188826068273, 0, 0, 1.38710966815627223188826068273, 1.73913013012394302611000660936, 3.01586743033391632798713887408, 3.90816121421174497149503325854, 4.08761638957052283837292033521, 4.40176178283255397633132059669, 5.20092372061562563148816864730, 5.40240999221953605980898689631, 6.31161855661158165100669513573, 6.63130700951211029382758832697, 6.88102877667591105738239412144, 7.00337385015735356382566383571, 7.973657625687070944670092759898, 8.406572554738207186652410747970, 8.724726512086160746890719913753, 8.794416419967053015032500806870, 9.523756629809606465296259622115, 10.19562869270034713479775873560

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