Properties

Label 4-224e2-1.1-c1e2-0-27
Degree $4$
Conductor $50176$
Sign $-1$
Analytic cond. $3.19926$
Root an. cond. $1.33740$
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  + 2·7-s − 6·9-s − 12·17-s − 6·25-s − 16·31-s + 4·41-s + 16·47-s + 3·49-s − 12·63-s + 16·71-s + 20·73-s − 32·79-s + 27·81-s − 12·89-s − 12·97-s + 32·103-s + 4·113-s − 24·119-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 72·153-s + 157-s + ⋯
L(s)  = 1  + 0.755·7-s − 2·9-s − 2.91·17-s − 6/5·25-s − 2.87·31-s + 0.624·41-s + 2.33·47-s + 3/7·49-s − 1.51·63-s + 1.89·71-s + 2.34·73-s − 3.60·79-s + 3·81-s − 1.27·89-s − 1.21·97-s + 3.15·103-s + 0.376·113-s − 2.20·119-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 5.82·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50176\)    =    \(2^{10} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(3.19926\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 50176,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−9.688287404638284990728895668207, −9.117894034638208548514808203742, −8.749304120934004278681759852831, −8.560607806232820418165457845354, −7.79292083571821938290976650763, −7.26382021666693896035142333427, −6.67949194679031572918581545842, −5.82613210142107824134972732669, −5.69394971603405508863657314834, −4.92225264716551837532790929786, −4.18465932715971087267449515138, −3.60173413237973468362473708205, −2.40145156924384381758797773878, −2.14556791802447766709825303506, 0, 2.14556791802447766709825303506, 2.40145156924384381758797773878, 3.60173413237973468362473708205, 4.18465932715971087267449515138, 4.92225264716551837532790929786, 5.69394971603405508863657314834, 5.82613210142107824134972732669, 6.67949194679031572918581545842, 7.26382021666693896035142333427, 7.79292083571821938290976650763, 8.560607806232820418165457845354, 8.749304120934004278681759852831, 9.117894034638208548514808203742, 9.688287404638284990728895668207

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