Properties

Label 4-546e2-1.1-c1e2-0-48
Degree $4$
Conductor $298116$
Sign $-1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 4·7-s − 4·8-s + 9-s − 8·11-s − 8·14-s + 5·16-s − 2·18-s + 16·22-s − 6·25-s + 12·28-s + 12·29-s − 6·32-s + 3·36-s − 4·37-s + 8·43-s − 24·44-s + 9·49-s + 12·50-s − 20·53-s − 16·56-s − 24·58-s + 4·63-s + 7·64-s − 32·67-s − 16·71-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.51·7-s − 1.41·8-s + 1/3·9-s − 2.41·11-s − 2.13·14-s + 5/4·16-s − 0.471·18-s + 3.41·22-s − 6/5·25-s + 2.26·28-s + 2.22·29-s − 1.06·32-s + 1/2·36-s − 0.657·37-s + 1.21·43-s − 3.61·44-s + 9/7·49-s + 1.69·50-s − 2.74·53-s − 2.13·56-s − 3.15·58-s + 0.503·63-s + 7/8·64-s − 3.90·67-s − 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−8.724491555695013698121297530874, −7.959290492018666343008631063702, −7.70223650354535578507572334847, −7.69854623831519517158383165327, −7.03828115713215538581036348773, −6.10664918750757402239876678098, −6.00899398361409811106287665786, −5.02692913607076442515975122964, −4.91686329817785172521042266230, −4.24383044940361627426015322662, −3.09122967370555540768759817180, −2.67335351129804876593036678201, −1.96145678743744747791090034638, −1.29963350151887452691131637880, 0, 1.29963350151887452691131637880, 1.96145678743744747791090034638, 2.67335351129804876593036678201, 3.09122967370555540768759817180, 4.24383044940361627426015322662, 4.91686329817785172521042266230, 5.02692913607076442515975122964, 6.00899398361409811106287665786, 6.10664918750757402239876678098, 7.03828115713215538581036348773, 7.69854623831519517158383165327, 7.70223650354535578507572334847, 7.959290492018666343008631063702, 8.724491555695013698121297530874

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