Properties

Label 4-84e4-1.1-c1e2-0-13
Degree $4$
Conductor $49787136$
Sign $1$
Analytic cond. $3174.47$
Root an. cond. $7.50616$
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  − 4·19-s + 7·25-s − 18·29-s + 10·31-s + 20·37-s − 24·47-s + 18·53-s + 18·59-s − 6·83-s − 8·103-s − 8·109-s − 12·113-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 0.917·19-s + 7/5·25-s − 3.34·29-s + 1.79·31-s + 3.28·37-s − 3.50·47-s + 2.47·53-s + 2.34·59-s − 0.658·83-s − 0.788·103-s − 0.766·109-s − 1.12·113-s + 1.72·121-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 + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(49787136\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3174.47\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7056} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 49787136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.079501099\)
\(L(\frac12)\) \(\approx\) \(3.079501099\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 166 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−8.167720925401891046948706943258, −7.86137978480158995060689498092, −7.23328218676107066828085211221, −7.21004650151500801986709774532, −6.67190983449897155422949657068, −6.43784893011807652739358521736, −6.00949109911626058410884031094, −5.69289053324231869194313502211, −5.26864435459295427135276181266, −5.01991687078121987870403959996, −4.35256486651412046606543450964, −4.23348982272405747351825015179, −3.91402477584844330865456522900, −3.30968187324325453840416597687, −2.84277555227717277885340193756, −2.63806026196075987504731752008, −1.83920404413216069526400203015, −1.82481290133913226245253154788, −0.816983947327139113656875989110, −0.55657533732187914545903509249, 0.55657533732187914545903509249, 0.816983947327139113656875989110, 1.82481290133913226245253154788, 1.83920404413216069526400203015, 2.63806026196075987504731752008, 2.84277555227717277885340193756, 3.30968187324325453840416597687, 3.91402477584844330865456522900, 4.23348982272405747351825015179, 4.35256486651412046606543450964, 5.01991687078121987870403959996, 5.26864435459295427135276181266, 5.69289053324231869194313502211, 6.00949109911626058410884031094, 6.43784893011807652739358521736, 6.67190983449897155422949657068, 7.21004650151500801986709774532, 7.23328218676107066828085211221, 7.86137978480158995060689498092, 8.167720925401891046948706943258

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