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Properties

Label 4-8788-1.1-c1e2-0-0
Degree $4$
Conductor $8788$
Sign $-1$
Analytic cond. $0.560330$
Root an. cond. $0.865189$
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  − 6·3-s + 4-s + 21·9-s − 6·12-s − 13-s + 16-s − 6·17-s − 8·23-s − 9·25-s − 54·27-s + 4·29-s + 21·36-s + 6·39-s − 10·43-s − 6·48-s − 13·49-s + 36·51-s − 52-s + 24·53-s − 16·61-s + 64-s − 6·68-s + 48·69-s + 54·75-s − 8·79-s + 108·81-s − 24·87-s + ⋯
L(s)  = 1  − 3.46·3-s + 1/2·4-s + 7·9-s − 1.73·12-s − 0.277·13-s + 1/4·16-s − 1.45·17-s − 1.66·23-s − 9/5·25-s − 10.3·27-s + 0.742·29-s + 7/2·36-s + 0.960·39-s − 1.52·43-s − 0.866·48-s − 1.85·49-s + 5.04·51-s − 0.138·52-s + 3.29·53-s − 2.04·61-s + 1/8·64-s − 0.727·68-s + 5.77·69-s + 6.23·75-s − 0.900·79-s + 12·81-s − 2.57·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8788\)    =    \(2^{2} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(0.560330\)
Root analytic conductor: \(0.865189\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 8788,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−11.49712673239204411601833482255, −11.04319412119313584483866781888, −10.28025762974353066563394220777, −10.24325458310862263428196114864, −9.539278397314208258267879412937, −8.310061634639968993764403624165, −7.45985560667588965978165906144, −6.79038227196567694631715258566, −6.40941752007862407532170431759, −5.88908450915949561478136129218, −5.41501572629242590776107371268, −4.61153453491182141362175201622, −4.09964629322041262514106195672, −1.82604108038319412812644968804, 0, 1.82604108038319412812644968804, 4.09964629322041262514106195672, 4.61153453491182141362175201622, 5.41501572629242590776107371268, 5.88908450915949561478136129218, 6.40941752007862407532170431759, 6.79038227196567694631715258566, 7.45985560667588965978165906144, 8.310061634639968993764403624165, 9.539278397314208258267879412937, 10.24325458310862263428196114864, 10.28025762974353066563394220777, 11.04319412119313584483866781888, 11.49712673239204411601833482255

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