Properties

Label 4-1492992-1.1-c1e2-0-20
Degree $4$
Conductor $1492992$
Sign $-1$
Analytic cond. $95.1944$
Root an. cond. $3.12358$
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  + 3-s + 9-s − 2·17-s + 25-s + 27-s − 18·41-s − 6·43-s + 11·49-s − 2·51-s − 12·59-s + 21·67-s + 3·73-s + 75-s + 81-s − 6·83-s − 26·97-s − 12·107-s + 20·113-s − 18·121-s − 18·123-s + 127-s − 6·129-s + 131-s + 137-s + 139-s + 11·147-s + 149-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.485·17-s + 1/5·25-s + 0.192·27-s − 2.81·41-s − 0.914·43-s + 11/7·49-s − 0.280·51-s − 1.56·59-s + 2.56·67-s + 0.351·73-s + 0.115·75-s + 1/9·81-s − 0.658·83-s − 2.63·97-s − 1.16·107-s + 1.88·113-s − 1.63·121-s − 1.62·123-s + 0.0887·127-s − 0.528·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.907·147-s + 0.0819·149-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: Trivial
Primitive: yes
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$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
29$C_2^2$ \( 1 + 27 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−7.81954433787047912420130843770, −7.14276203698108987421908744348, −6.87083409016250175547768290582, −6.59223496572461597345140876091, −5.99827839764525534934648476274, −5.35719608952719231907701493400, −5.10333982085315791055604865036, −4.54099316603993926253598227078, −3.98286943701874096398638222736, −3.56295147327172625200245118469, −3.03473364432633697782543300813, −2.45273194171846368113264877012, −1.86758477823898263645962333276, −1.21131495596851593801387590776, 0, 1.21131495596851593801387590776, 1.86758477823898263645962333276, 2.45273194171846368113264877012, 3.03473364432633697782543300813, 3.56295147327172625200245118469, 3.98286943701874096398638222736, 4.54099316603993926253598227078, 5.10333982085315791055604865036, 5.35719608952719231907701493400, 5.99827839764525534934648476274, 6.59223496572461597345140876091, 6.87083409016250175547768290582, 7.14276203698108987421908744348, 7.81954433787047912420130843770

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