Properties

Label 4-38e2-1.1-c2e2-0-0
Degree $4$
Conductor $1444$
Sign $1$
Analytic cond. $1.07210$
Root an. cond. $1.01755$
Motivic weight $2$
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  − 2·4-s − 2·5-s + 10·7-s + 10·9-s + 10·11-s + 4·16-s − 50·17-s + 38·19-s + 4·20-s − 20·23-s − 47·25-s − 20·28-s − 20·35-s − 20·36-s + 10·43-s − 20·44-s − 20·45-s + 10·47-s − 23·49-s − 20·55-s + 190·61-s + 100·63-s − 8·64-s + 100·68-s − 50·73-s − 76·76-s + 100·77-s + ⋯
L(s)  = 1  − 1/2·4-s − 2/5·5-s + 10/7·7-s + 10/9·9-s + 0.909·11-s + 1/4·16-s − 2.94·17-s + 2·19-s + 1/5·20-s − 0.869·23-s − 1.87·25-s − 5/7·28-s − 4/7·35-s − 5/9·36-s + 0.232·43-s − 0.454·44-s − 4/9·45-s + 0.212·47-s − 0.469·49-s − 0.363·55-s + 3.11·61-s + 1.58·63-s − 1/8·64-s + 1.47·68-s − 0.684·73-s − 76-s + 1.29·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1.07210\)
Root analytic conductor: \(1.01755\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1444,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.081515681\)
\(L(\frac12)\) \(\approx\) \(1.081515681\)
\(L(2)\) 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$C_2$ \( 1 + p T^{2} \)
19$C_1$ \( ( 1 - p T )^{2} \)
good3$C_2^2$ \( 1 - 10 T^{2} + p^{4} T^{4} \)
5$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 50 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 118 T^{2} + p^{4} T^{4} \)
31$C_2^2$ \( 1 - 122 T^{2} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 2090 T^{2} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 1562 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
47$C_2$ \( ( 1 - 5 T + p^{2} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 4970 T^{2} + p^{4} T^{4} \)
59$C_2$ \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \)
61$C_2$ \( ( 1 - 95 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 3190 T^{2} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 25 T + p^{2} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10682 T^{2} + p^{4} T^{4} \)
83$C_2$ \( ( 1 + 130 T + p^{2} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 358 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 18530 T^{2} + p^{4} T^{4} \)
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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

−16.04461245024572773910857014457, −15.77567872196033378069628506493, −15.41560992737907837824552420985, −14.43117277388627112367016167710, −14.16627309687701457421252395232, −13.39084494492758226668165550448, −13.04614192462643548758131361985, −11.98348497932857562733541939298, −11.35769899751648216775133019409, −11.35388525350450538182918253350, −10.08113353934198064649439268782, −9.568098904237598978724782117449, −8.785817704554103179688929946246, −8.110799446618265920768383417371, −7.38922964383083705987277679191, −6.66675494854769332319275468455, −5.44173856762884846101821411921, −4.39427648108079931839440110889, −4.05897326619392014022796515472, −1.80246226895369231409882706105, 1.80246226895369231409882706105, 4.05897326619392014022796515472, 4.39427648108079931839440110889, 5.44173856762884846101821411921, 6.66675494854769332319275468455, 7.38922964383083705987277679191, 8.110799446618265920768383417371, 8.785817704554103179688929946246, 9.568098904237598978724782117449, 10.08113353934198064649439268782, 11.35388525350450538182918253350, 11.35769899751648216775133019409, 11.98348497932857562733541939298, 13.04614192462643548758131361985, 13.39084494492758226668165550448, 14.16627309687701457421252395232, 14.43117277388627112367016167710, 15.41560992737907837824552420985, 15.77567872196033378069628506493, 16.04461245024572773910857014457

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