Properties

Label 4-2736e2-1.1-c2e2-0-4
Degree $4$
Conductor $7485696$
Sign $1$
Analytic cond. $5557.79$
Root an. cond. $8.63426$
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 − 3·11-s + 15·17-s + 38·19-s − 60·23-s + 25·25-s + 45·35-s − 85·43-s − 75·47-s + 49·49-s + 27·55-s − 103·61-s + 25·73-s + 15·77-s + 180·83-s − 135·85-s − 342·95-s + 204·101-s + 540·115-s − 75·119-s + 121·121-s + 54·125-s + 127-s + 131-s − 190·133-s + 137-s + ⋯
L(s)  = 1  − 9/5·5-s − 5/7·7-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 − 1.59·47-s + 49-s + 0.490·55-s − 1.68·61-s + 0.342·73-s + 0.194·77-s + 2.16·83-s − 1.58·85-s − 3.59·95-s + 2.01·101-s + 4.69·115-s − 0.630·119-s + 121-s + 0.431·125-s + 0.00787·127-s + 0.00763·131-s − 1.42·133-s + 0.00729·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7485696\)    =    \(2^{8} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5557.79\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7485696,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4877877894\)
\(L(\frac12)\) \(\approx\) \(0.4877877894\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
19$C_1$ \( ( 1 - p T )^{2} \)
good5$C_2^2$ \( 1 + 9 T + 56 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \)
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

−8.623196319287837903710861543735, −8.409433514280162903342807335070, −7.84766702032665860812898650828, −7.78468708734004596451211640617, −7.51683994690503317021303142135, −7.13188712623525053583645225339, −6.44946638933083919877005524054, −6.30792460053435373934037708723, −5.70521743749958004090292382849, −5.41662330633094342821036707430, −4.76468423156670947018962824839, −4.59041093401239313367140045706, −3.81948559759058607693665009804, −3.64629273637404745869130457410, −3.29777410413099561595187704300, −2.99909277442492232223557999549, −2.13122823598996888680509362040, −1.61980592478596053074992483231, −0.809721063769823732680920013632, −0.21634792284992863637742898794, 0.21634792284992863637742898794, 0.809721063769823732680920013632, 1.61980592478596053074992483231, 2.13122823598996888680509362040, 2.99909277442492232223557999549, 3.29777410413099561595187704300, 3.64629273637404745869130457410, 3.81948559759058607693665009804, 4.59041093401239313367140045706, 4.76468423156670947018962824839, 5.41662330633094342821036707430, 5.70521743749958004090292382849, 6.30792460053435373934037708723, 6.44946638933083919877005524054, 7.13188712623525053583645225339, 7.51683994690503317021303142135, 7.78468708734004596451211640617, 7.84766702032665860812898650828, 8.409433514280162903342807335070, 8.623196319287837903710861543735

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