Properties

Label 4-1492992-1.1-c1e2-0-4
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·11-s − 8·17-s + 12·19-s − 9·25-s + 20·41-s + 12·43-s − 5·49-s + 24·59-s + 12·67-s + 18·73-s + 6·83-s − 28·89-s − 18·97-s + 30·107-s + 20·113-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + ⋯
L(s)  = 1  + 1.80·11-s − 1.94·17-s + 2.75·19-s − 9/5·25-s + 3.12·41-s + 1.82·43-s − 5/7·49-s + 3.12·59-s + 1.46·67-s + 2.10·73-s + 0.658·83-s − 2.96·89-s − 1.82·97-s + 2.90·107-s + 1.88·113-s + 5/11·121-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 − 2·169-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: \(0\)
Selberg data: \((4,\ 1492992,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.671552123\)
\(L(\frac12)\) \(\approx\) \(2.671552123\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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

−7.75120135592916568614052087932, −7.47224706156871722823044931973, −7.07642279360820371836072015909, −6.65325213254225917768927617170, −6.14570823384352111267011079375, −5.81482520780945663550334444725, −5.33767563665680457913764465863, −4.78635677042327797457152601343, −4.07336899816643905031985228211, −3.94091986861202608778869492074, −3.52624329017255126054742371711, −2.42238256260612007081644598630, −2.39632866530783962583020681104, −1.30229225574123782841215729642, −0.794899488559166432876984150469, 0.794899488559166432876984150469, 1.30229225574123782841215729642, 2.39632866530783962583020681104, 2.42238256260612007081644598630, 3.52624329017255126054742371711, 3.94091986861202608778869492074, 4.07336899816643905031985228211, 4.78635677042327797457152601343, 5.33767563665680457913764465863, 5.81482520780945663550334444725, 6.14570823384352111267011079375, 6.65325213254225917768927617170, 7.07642279360820371836072015909, 7.47224706156871722823044931973, 7.75120135592916568614052087932

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