Properties

Label 4-1134e2-1.1-c1e2-0-49
Degree $4$
Conductor $1285956$
Sign $-1$
Analytic cond. $81.9936$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·7-s + 16-s − 10·25-s + 2·28-s + 8·37-s + 2·43-s − 3·49-s − 64-s + 10·67-s − 8·79-s + 10·100-s + 32·109-s − 2·112-s + 13·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 2·172-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.755·7-s + 1/4·16-s − 2·25-s + 0.377·28-s + 1.31·37-s + 0.304·43-s − 3/7·49-s − 1/8·64-s + 1.22·67-s − 0.900·79-s + 100-s + 3.06·109-s − 0.188·112-s + 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-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 − 0.152·172-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1285956\)    =    \(2^{2} \cdot 3^{8} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(81.9936\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1285956,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3 \( 1 \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−7.79148805932115420247707092925, −7.36590587051883176588162902755, −6.96804654506642829427764156507, −6.33188547826714627564906416868, −5.98480672295556885636255944174, −5.72456645987052214307087784093, −5.09210178025586626300727285827, −4.51510535925306464450190785265, −4.17083629203731894621799547427, −3.55547551391661045042903762903, −3.23471626636013325843638519626, −2.43915291213484878909906296965, −1.92982270068984898185712881030, −0.940622612759610308580341061216, 0, 0.940622612759610308580341061216, 1.92982270068984898185712881030, 2.43915291213484878909906296965, 3.23471626636013325843638519626, 3.55547551391661045042903762903, 4.17083629203731894621799547427, 4.51510535925306464450190785265, 5.09210178025586626300727285827, 5.72456645987052214307087784093, 5.98480672295556885636255944174, 6.33188547826714627564906416868, 6.96804654506642829427764156507, 7.36590587051883176588162902755, 7.79148805932115420247707092925

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