Properties

Label 4-1152e2-1.1-c1e2-0-20
Degree $4$
Conductor $1327104$
Sign $1$
Analytic cond. $84.6173$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·9-s + 5·11-s + 2·17-s + 4·19-s + 3·25-s + 5·27-s − 5·33-s + 9·41-s + 9·43-s + 11·49-s − 2·51-s − 4·57-s − 5·59-s − 67-s + 22·73-s − 3·75-s + 81-s − 18·83-s − 4·89-s − 7·97-s − 10·99-s + 20·107-s + 10·113-s − 121-s − 9·123-s + 127-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s + 1.50·11-s + 0.485·17-s + 0.917·19-s + 3/5·25-s + 0.962·27-s − 0.870·33-s + 1.40·41-s + 1.37·43-s + 11/7·49-s − 0.280·51-s − 0.529·57-s − 0.650·59-s − 0.122·67-s + 2.57·73-s − 0.346·75-s + 1/9·81-s − 1.97·83-s − 0.423·89-s − 0.710·97-s − 1.00·99-s + 1.93·107-s + 0.940·113-s − 0.0909·121-s − 0.811·123-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1327104\)    =    \(2^{14} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(84.6173\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1327104,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.043291702\)
\(L(\frac12)\) \(\approx\) \(2.043291702\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \)
13$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 51 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 145 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−7.962351663940697687149320631915, −7.44061248368216038821195196858, −7.08967655161547763312491745850, −6.62828446204442263097407540051, −6.19553576499110461544104173717, −5.68311411805138590008789276208, −5.55959233826699695884627212754, −4.87565888376787101902271473648, −4.34012925683234858809191178500, −3.93306953188418835977176231410, −3.33503453095546112372983058239, −2.82158829229719479933337466458, −2.18718694046518648525685563337, −1.20076695803606577699683124223, −0.77066389391977975295814049936, 0.77066389391977975295814049936, 1.20076695803606577699683124223, 2.18718694046518648525685563337, 2.82158829229719479933337466458, 3.33503453095546112372983058239, 3.93306953188418835977176231410, 4.34012925683234858809191178500, 4.87565888376787101902271473648, 5.55959233826699695884627212754, 5.68311411805138590008789276208, 6.19553576499110461544104173717, 6.62828446204442263097407540051, 7.08967655161547763312491745850, 7.44061248368216038821195196858, 7.962351663940697687149320631915

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