Properties

Label 4-112e3-1.1-c1e2-0-5
Degree $4$
Conductor $1404928$
Sign $1$
Analytic cond. $89.5794$
Root an. cond. $3.07646$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 2·9-s + 4·11-s + 4·23-s − 2·25-s − 6·29-s + 2·37-s + 4·43-s + 49-s + 18·53-s − 2·63-s + 12·67-s + 12·71-s + 4·77-s + 12·79-s − 5·81-s − 8·99-s + 20·107-s − 22·109-s + 4·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.377·7-s − 2/3·9-s + 1.20·11-s + 0.834·23-s − 2/5·25-s − 1.11·29-s + 0.328·37-s + 0.609·43-s + 1/7·49-s + 2.47·53-s − 0.251·63-s + 1.46·67-s + 1.42·71-s + 0.455·77-s + 1.35·79-s − 5/9·81-s − 0.804·99-s + 1.93·107-s − 2.10·109-s + 0.376·113-s + 6/11·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1404928\)    =    \(2^{12} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(89.5794\)
Root analytic conductor: \(3.07646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1404928,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.283229226\)
\(L(\frac12)\) \(\approx\) \(2.283229226\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( 1 - T \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
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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

−8.034666256708262515824866917046, −7.40559669902683393640507717789, −7.12892461900060831969985568807, −6.68665122667224520758572588307, −6.14241293746386195154364597896, −5.81331355743005549651380911316, −5.26614892127252096312002523037, −4.94905534492352257320314762240, −4.25419096687139986858488692501, −3.69158807981670404645914133292, −3.59934917140019282794104459612, −2.58252216631041919756131247572, −2.26340832547008833080124346844, −1.41517817445428925846563194836, −0.69853749616174011802263236645, 0.69853749616174011802263236645, 1.41517817445428925846563194836, 2.26340832547008833080124346844, 2.58252216631041919756131247572, 3.59934917140019282794104459612, 3.69158807981670404645914133292, 4.25419096687139986858488692501, 4.94905534492352257320314762240, 5.26614892127252096312002523037, 5.81331355743005549651380911316, 6.14241293746386195154364597896, 6.68665122667224520758572588307, 7.12892461900060831969985568807, 7.40559669902683393640507717789, 8.034666256708262515824866917046

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