Properties

Label 4-1216e2-1.1-c2e2-0-7
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $1097.83$
Root an. cond. $5.75617$
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  − 9·5-s − 5·7-s + 18·9-s − 3·11-s − 15·17-s − 38·19-s + 60·23-s + 25·25-s + 45·35-s + 85·43-s − 162·45-s + 75·47-s + 49·49-s + 27·55-s + 103·61-s − 90·63-s + 25·73-s + 15·77-s + 243·81-s + 180·83-s + 135·85-s + 342·95-s − 54·99-s + 204·101-s − 540·115-s + 75·119-s + 121·121-s + ⋯
L(s)  = 1  − 9/5·5-s − 5/7·7-s + 2·9-s − 0.272·11-s − 0.882·17-s − 2·19-s + 2.60·23-s + 25-s + 9/7·35-s + 1.97·43-s − 3.59·45-s + 1.59·47-s + 49-s + 0.490·55-s + 1.68·61-s − 1.42·63-s + 0.342·73-s + 0.194·77-s + 3·81-s + 2.16·83-s + 1.58·85-s + 18/5·95-s − 0.545·99-s + 2.01·101-s − 4.69·115-s + 0.630·119-s + 121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1097.83\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.987505834\)
\(L(\frac12)\) \(\approx\) \(1.987505834\)
\(L(2)\) 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 \( 1 \)
19$C_1$ \( ( 1 + p T )^{2} \)
good3$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
5$C_2^2$ \( 1 + 9 T + 56 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \)
7$C_2^2$ \( 1 + 5 T - 24 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 112 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
17$C_2^2$ \( 1 + 15 T - 64 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_2^2$ \( 1 - 85 T + 5376 T^{2} - 85 p^{2} T^{3} + p^{4} T^{4} \)
47$C_2^2$ \( 1 - 75 T + 3416 T^{2} - 75 p^{2} T^{3} + p^{4} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2^2$ \( 1 - 103 T + 6888 T^{2} - 103 p^{2} T^{3} + p^{4} T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 - 25 T - 4704 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
83$C_2$ \( ( 1 - 90 T + p^{2} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
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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

−9.684623603709311220993048002702, −9.249072203508443292901674703241, −8.870346499706030835317205719410, −8.663909534111799877263841375591, −7.959378767688993249294202391933, −7.65173385102511865988847372984, −7.20835440471280929062239227491, −7.00112076372468266461236978756, −6.63048977005336961018261782124, −6.17953315716183805416019082257, −5.44265823401511124272764364440, −4.82075029923866778757549980192, −4.33692293712700051648747847624, −4.23208177294235247869460295025, −3.74204118672950998808912107182, −3.26839704221711211886242645510, −2.44084694335786834135390808550, −2.03830946800499647329407192931, −0.840945592279295720297140119457, −0.59270106204590569644357486746, 0.59270106204590569644357486746, 0.840945592279295720297140119457, 2.03830946800499647329407192931, 2.44084694335786834135390808550, 3.26839704221711211886242645510, 3.74204118672950998808912107182, 4.23208177294235247869460295025, 4.33692293712700051648747847624, 4.82075029923866778757549980192, 5.44265823401511124272764364440, 6.17953315716183805416019082257, 6.63048977005336961018261782124, 7.00112076372468266461236978756, 7.20835440471280929062239227491, 7.65173385102511865988847372984, 7.959378767688993249294202391933, 8.663909534111799877263841375591, 8.870346499706030835317205719410, 9.249072203508443292901674703241, 9.684623603709311220993048002702

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