Properties

Label 4-912e2-1.1-c1e2-0-4
Degree $4$
Conductor $831744$
Sign $1$
Analytic cond. $53.0327$
Root an. cond. $2.69858$
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  − 8·7-s − 3·9-s + 8·19-s + 10·25-s − 16·43-s + 34·49-s − 28·61-s + 24·63-s − 20·73-s + 9·81-s + 22·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 24·171-s + 173-s − 80·175-s + 179-s + 181-s + ⋯
L(s)  = 1  − 3.02·7-s − 9-s + 1.83·19-s + 2·25-s − 2.43·43-s + 34/7·49-s − 3.58·61-s + 3.02·63-s − 2.34·73-s + 81-s + 2·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s − 1.83·171-s + 0.0760·173-s − 6.04·175-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(831744\)    =    \(2^{8} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(53.0327\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 831744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5859410580\)
\(L(\frac12)\) \(\approx\) \(0.5859410580\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
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

−10.16761401018975576072128546912, −9.960644255468066515084109212125, −9.351770335484150192455132885692, −9.204770062142631447508928788500, −8.769787722509297637281614053890, −8.412089248664779769199288113932, −7.45556470841348436466321618004, −7.43590336353570720291134580689, −6.76865001662113966146381760546, −6.30212896485923721085145632104, −6.25752534180307998816673657423, −5.59665502603094405824843497903, −5.12799861132040569122094833053, −4.59908814655601506373740050463, −3.69832705049264906205962252807, −3.21295115422570410239520885970, −3.01966082202961335467321198093, −2.76664296321127246458074039600, −1.43059509108331842047903431619, −0.36983345952922080602492014411, 0.36983345952922080602492014411, 1.43059509108331842047903431619, 2.76664296321127246458074039600, 3.01966082202961335467321198093, 3.21295115422570410239520885970, 3.69832705049264906205962252807, 4.59908814655601506373740050463, 5.12799861132040569122094833053, 5.59665502603094405824843497903, 6.25752534180307998816673657423, 6.30212896485923721085145632104, 6.76865001662113966146381760546, 7.43590336353570720291134580689, 7.45556470841348436466321618004, 8.412089248664779769199288113932, 8.769787722509297637281614053890, 9.204770062142631447508928788500, 9.351770335484150192455132885692, 9.960644255468066515084109212125, 10.16761401018975576072128546912

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