Properties

Label 4-1492992-1.1-c1e2-0-26
Degree $4$
Conductor $1492992$
Sign $-1$
Analytic cond. $95.1944$
Root an. cond. $3.12358$
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  + 12·11-s − 4·17-s − 6·19-s − 6·25-s + 16·41-s − 24·43-s − 5·49-s + 12·59-s − 6·67-s − 30·73-s − 24·83-s − 20·89-s + 18·97-s − 12·107-s + 28·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 17·169-s + 173-s + ⋯
L(s)  = 1  + 3.61·11-s − 0.970·17-s − 1.37·19-s − 6/5·25-s + 2.49·41-s − 3.65·43-s − 5/7·49-s + 1.56·59-s − 0.733·67-s − 3.51·73-s − 2.63·83-s − 2.11·89-s + 1.82·97-s − 1.16·107-s + 2.63·113-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-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 − 1.30·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1492992\)    =    \(2^{11} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(95.1944\)
Root analytic conductor: \(3.12358\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1492992} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1492992,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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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

−7.53716928854145246282578201442, −7.17330313041166053658325455178, −6.70247851531082591461292587351, −6.47142445887211239233670215635, −6.03993649285615255926110154202, −5.80426002862581950962542400694, −4.81872092436289835248079798823, −4.42534002049006891974002112914, −3.95165764595971724881843530149, −3.89073820849432411300555838274, −3.12135221332905635448639751702, −2.32639594714730545418397709938, −1.55247314198910365347987668608, −1.39920923199288070248453149386, 0, 1.39920923199288070248453149386, 1.55247314198910365347987668608, 2.32639594714730545418397709938, 3.12135221332905635448639751702, 3.89073820849432411300555838274, 3.95165764595971724881843530149, 4.42534002049006891974002112914, 4.81872092436289835248079798823, 5.80426002862581950962542400694, 6.03993649285615255926110154202, 6.47142445887211239233670215635, 6.70247851531082591461292587351, 7.17330313041166053658325455178, 7.53716928854145246282578201442

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