Properties

Label 4-912e2-1.1-c1e2-0-102
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s − 10·11-s − 10·17-s + 2·19-s − 25-s − 4·27-s + 20·33-s − 16·41-s + 18·43-s − 13·49-s + 20·51-s − 4·57-s − 16·59-s − 22·73-s + 2·75-s + 5·81-s − 8·83-s + 20·89-s − 20·97-s − 30·99-s + 4·107-s − 28·113-s + 53·121-s + 32·123-s + 127-s − 36·129-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 3.01·11-s − 2.42·17-s + 0.458·19-s − 1/5·25-s − 0.769·27-s + 3.48·33-s − 2.49·41-s + 2.74·43-s − 1.85·49-s + 2.80·51-s − 0.529·57-s − 2.08·59-s − 2.57·73-s + 0.230·75-s + 5/9·81-s − 0.878·83-s + 2.11·89-s − 2.03·97-s − 3.01·99-s + 0.386·107-s − 2.63·113-s + 4.81·121-s + 2.88·123-s + 0.0887·127-s − 3.16·129-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: \(2\)
Selberg data: \((4,\ 831744,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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

−7.64895552693460443584303369212, −7.37837081889035825525626443160, −6.89871092668857764939078364279, −6.32311719558750517700420901480, −5.99556993924043936846063270863, −5.39401606702244685695168285894, −5.13360434530387425158153694109, −4.51287023030762779757348110603, −4.46915881587796642630977597343, −3.40954725176495870801828644718, −2.74155671051499470855999521696, −2.33981183738947215489783359267, −1.54824828965757383971408957925, 0, 0, 1.54824828965757383971408957925, 2.33981183738947215489783359267, 2.74155671051499470855999521696, 3.40954725176495870801828644718, 4.46915881587796642630977597343, 4.51287023030762779757348110603, 5.13360434530387425158153694109, 5.39401606702244685695168285894, 5.99556993924043936846063270863, 6.32311719558750517700420901480, 6.89871092668857764939078364279, 7.37837081889035825525626443160, 7.64895552693460443584303369212

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