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Properties

Label 4-88e2-1.1-c1e2-0-0
Degree $4$
Conductor $7744$
Sign $-1$
Analytic cond. $0.493764$
Root an. cond. $0.838262$
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  − 2·2-s − 2·3-s + 2·4-s + 4·6-s − 3·9-s + 2·11-s − 4·12-s − 4·16-s − 4·17-s + 6·18-s − 4·22-s − 9·25-s + 14·27-s + 8·32-s − 4·33-s + 8·34-s − 6·36-s − 16·41-s − 12·43-s + 4·44-s + 8·48-s − 10·49-s + 18·50-s + 8·51-s − 28·54-s + 10·59-s − 8·64-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s − 9-s + 0.603·11-s − 1.15·12-s − 16-s − 0.970·17-s + 1.41·18-s − 0.852·22-s − 9/5·25-s + 2.69·27-s + 1.41·32-s − 0.696·33-s + 1.37·34-s − 36-s − 2.49·41-s − 1.82·43-s + 0.603·44-s + 1.15·48-s − 1.42·49-s + 2.54·50-s + 1.12·51-s − 3.81·54-s + 1.30·59-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7744\)    =    \(2^{6} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(0.493764\)
Root analytic conductor: \(0.838262\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 7744,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−11.45125861034521058353374083886, −11.04905759572316236285673661023, −10.14466147519938939509327606244, −10.03550909718107888433464868208, −9.092146666164135933217503972545, −8.603539619290756001226038948684, −8.248674172646585313129242503450, −7.39701457500400876162065539217, −6.46350440936163119026889680667, −6.36261389471308870138602900888, −5.32992026591659065835464612635, −4.69659420732828154227208765880, −3.40579821695092674516778014630, −1.94547887078347189383086322028, 0, 1.94547887078347189383086322028, 3.40579821695092674516778014630, 4.69659420732828154227208765880, 5.32992026591659065835464612635, 6.36261389471308870138602900888, 6.46350440936163119026889680667, 7.39701457500400876162065539217, 8.248674172646585313129242503450, 8.603539619290756001226038948684, 9.092146666164135933217503972545, 10.03550909718107888433464868208, 10.14466147519938939509327606244, 11.04905759572316236285673661023, 11.45125861034521058353374083886

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