Properties

Label 4-2014e2-1.1-c1e2-0-0
Degree $4$
Conductor $4056196$
Sign $1$
Analytic cond. $258.626$
Root an. cond. $4.01022$
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  + 4-s − 2·7-s − 5·9-s − 12·11-s + 10·13-s + 16-s + 6·17-s − 10·25-s − 2·28-s + 18·29-s − 5·36-s + 4·37-s + 16·43-s − 12·44-s − 11·49-s + 10·52-s − 3·53-s + 18·59-s + 10·63-s + 64-s + 6·68-s + 24·77-s + 16·81-s − 24·89-s − 20·91-s − 20·97-s + 60·99-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.755·7-s − 5/3·9-s − 3.61·11-s + 2.77·13-s + 1/4·16-s + 1.45·17-s − 2·25-s − 0.377·28-s + 3.34·29-s − 5/6·36-s + 0.657·37-s + 2.43·43-s − 1.80·44-s − 1.57·49-s + 1.38·52-s − 0.412·53-s + 2.34·59-s + 1.25·63-s + 1/8·64-s + 0.727·68-s + 2.73·77-s + 16/9·81-s − 2.54·89-s − 2.09·91-s − 2.03·97-s + 6.03·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4056196\)    =    \(2^{2} \cdot 19^{2} \cdot 53^{2}\)
Sign: $1$
Analytic conductor: \(258.626\)
Root analytic conductor: \(4.01022\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4056196,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.472981981\)
\(L(\frac12)\) \(\approx\) \(1.472981981\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
53$C_2$ \( 1 + 3 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−7.73591843742988285704805588989, −6.91510043982367309787610822998, −6.47867494831462502206457077359, −5.97200219047894843621852288003, −5.81027585652647529822118508915, −5.48865938153480470174837316173, −5.25486930890297982669987800379, −4.41999424159624366152179632530, −3.91345987377005316159897465904, −3.19009856541377672983850265608, −2.99432139105097413830284496966, −2.71849777094368987092889592454, −2.21356991955325344340570102497, −1.14633070091081217096891416847, −0.47096046911660324297322742886, 0.47096046911660324297322742886, 1.14633070091081217096891416847, 2.21356991955325344340570102497, 2.71849777094368987092889592454, 2.99432139105097413830284496966, 3.19009856541377672983850265608, 3.91345987377005316159897465904, 4.41999424159624366152179632530, 5.25486930890297982669987800379, 5.48865938153480470174837316173, 5.81027585652647529822118508915, 5.97200219047894843621852288003, 6.47867494831462502206457077359, 6.91510043982367309787610822998, 7.73591843742988285704805588989

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