Properties

Label 4-616e2-1.1-c1e2-0-7
Degree $4$
Conductor $379456$
Sign $1$
Analytic cond. $24.1944$
Root an. cond. $2.21783$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 2·5-s − 2·7-s − 5·9-s − 4·10-s + 2·11-s + 8·13-s + 4·14-s − 4·16-s + 10·18-s + 4·20-s − 4·22-s − 7·25-s − 16·26-s − 4·28-s + 14·31-s + 8·32-s − 4·35-s − 10·36-s − 12·43-s + 4·44-s − 10·45-s + 16·47-s − 3·49-s + 14·50-s + 16·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.894·5-s − 0.755·7-s − 5/3·9-s − 1.26·10-s + 0.603·11-s + 2.21·13-s + 1.06·14-s − 16-s + 2.35·18-s + 0.894·20-s − 0.852·22-s − 7/5·25-s − 3.13·26-s − 0.755·28-s + 2.51·31-s + 1.41·32-s − 0.676·35-s − 5/3·36-s − 1.82·43-s + 0.603·44-s − 1.49·45-s + 2.33·47-s − 3/7·49-s + 1.97·50-s + 2.21·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(379456\)    =    \(2^{6} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(24.1944\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 379456,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8610581309\)
\(L(\frac12)\) \(\approx\) \(0.8610581309\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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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

−8.627153557857953172015491503977, −8.603539619290756001226038948684, −8.016470746894750837719454735098, −7.46973642361763444298598982747, −6.68253837392674763250909414891, −6.36261389471308870138602900888, −6.05856391223997850967309901807, −5.71143819833623906703502312707, −4.99767010522499384059415710151, −4.07790776750078242353026984533, −3.63865093449500368439278742924, −2.90247167943624077903906049478, −2.30747219091521399299623542576, −1.49733172832762767313391351767, −0.68419642869801904316651703930, 0.68419642869801904316651703930, 1.49733172832762767313391351767, 2.30747219091521399299623542576, 2.90247167943624077903906049478, 3.63865093449500368439278742924, 4.07790776750078242353026984533, 4.99767010522499384059415710151, 5.71143819833623906703502312707, 6.05856391223997850967309901807, 6.36261389471308870138602900888, 6.68253837392674763250909414891, 7.46973642361763444298598982747, 8.016470746894750837719454735098, 8.603539619290756001226038948684, 8.627153557857953172015491503977

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