Properties

Label 4-1368e2-1.1-c1e2-0-17
Degree $4$
Conductor $1871424$
Sign $1$
Analytic cond. $119.323$
Root an. cond. $3.30507$
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  + 3·5-s + 8·11-s + 5·13-s − 5·17-s + 8·19-s − 23-s + 5·25-s + 3·29-s + 8·31-s + 4·37-s − 5·41-s + 11·43-s − 5·47-s − 14·49-s − 9·53-s + 24·55-s + 13·59-s + 61-s + 15·65-s + 5·67-s + 71-s + 9·73-s − 17·79-s − 32·83-s − 15·85-s + 3·89-s + 24·95-s + ⋯
L(s)  = 1  + 1.34·5-s + 2.41·11-s + 1.38·13-s − 1.21·17-s + 1.83·19-s − 0.208·23-s + 25-s + 0.557·29-s + 1.43·31-s + 0.657·37-s − 0.780·41-s + 1.67·43-s − 0.729·47-s − 2·49-s − 1.23·53-s + 3.23·55-s + 1.69·59-s + 0.128·61-s + 1.86·65-s + 0.610·67-s + 0.118·71-s + 1.05·73-s − 1.91·79-s − 3.51·83-s − 1.62·85-s + 0.317·89-s + 2.46·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1871424\)    =    \(2^{6} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(119.323\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1871424,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.518147496\)
\(L(\frac12)\) \(\approx\) \(4.518147496\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2^2$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−9.749983469200895253463882489851, −9.367681820947984554783722188979, −8.991779058125508390947565010689, −8.828028912624420470127963299747, −8.248787913468227015034173976083, −7.928806002141256890635426187922, −7.11649782786393718343444923454, −6.80868988005879988007847041521, −6.34505723505851555093738568329, −6.29180913169881659201104257972, −5.77633603010819234277034443891, −5.30427868616545111350931758606, −4.51418192937962013112816625171, −4.44977473391486715702264048887, −3.64121172677040595013894719947, −3.32855668736504267473873279740, −2.65401172272635213634698606324, −1.95471061394571443070945452840, −1.19434886791347851552481812535, −1.16206996123835915376356632410, 1.16206996123835915376356632410, 1.19434886791347851552481812535, 1.95471061394571443070945452840, 2.65401172272635213634698606324, 3.32855668736504267473873279740, 3.64121172677040595013894719947, 4.44977473391486715702264048887, 4.51418192937962013112816625171, 5.30427868616545111350931758606, 5.77633603010819234277034443891, 6.29180913169881659201104257972, 6.34505723505851555093738568329, 6.80868988005879988007847041521, 7.11649782786393718343444923454, 7.928806002141256890635426187922, 8.248787913468227015034173976083, 8.828028912624420470127963299747, 8.991779058125508390947565010689, 9.367681820947984554783722188979, 9.749983469200895253463882489851

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