Properties

Label 4-15e4-1.1-c3e2-0-6
Degree $4$
Conductor $50625$
Sign $1$
Analytic cond. $176.237$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·4-s + 30·7-s + 70·13-s − 28·16-s + 182·19-s − 180·28-s − 294·31-s + 740·37-s + 670·43-s − 11·49-s − 420·52-s + 854·61-s + 552·64-s + 30·67-s − 140·73-s − 1.09e3·76-s − 1.75e3·79-s + 2.10e3·91-s − 2.17e3·97-s + 3.08e3·103-s − 1.62e3·109-s − 840·112-s + 1.33e3·121-s + 1.76e3·124-s + 127-s + 131-s + 5.46e3·133-s + ⋯
L(s)  = 1  − 3/4·4-s + 1.61·7-s + 1.49·13-s − 0.437·16-s + 2.19·19-s − 1.21·28-s − 1.70·31-s + 3.28·37-s + 2.37·43-s − 0.0320·49-s − 1.12·52-s + 1.79·61-s + 1.07·64-s + 0.0547·67-s − 0.224·73-s − 1.64·76-s − 2.49·79-s + 2.41·91-s − 2.27·97-s + 2.94·103-s − 1.42·109-s − 0.708·112-s + 1.00·121-s + 1.27·124-s + 0.000698·127-s + 0.000666·131-s + 3.55·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50625\)    =    \(3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(176.237\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 50625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.996371727\)
\(L(\frac12)\) \(\approx\) \(2.996371727\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
good2$C_2^2$ \( 1 + 3 p T^{2} + p^{6} T^{4} \)
7$C_2$ \( ( 1 - 15 T + p^{3} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 1338 T^{2} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 35 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 1986 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 91 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 11374 T^{2} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 44778 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 + 147 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 p T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 58158 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 - 335 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 176286 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 289914 T^{2} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 373242 T^{2} + p^{6} T^{4} \)
61$C_2$ \( ( 1 - 7 p T + p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 15 T + p^{3} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 711822 T^{2} + p^{6} T^{4} \)
73$C_2$ \( ( 1 + 70 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 876 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 861334 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 1085 T + p^{3} T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.79500268335814591671195664500, −11.40357111009650033936710953608, −11.06625406840793981912470562700, −11.01651792981626866405587340870, −9.789727895342154361174443507442, −9.740547522904466300445649163495, −8.890664913287295193998741414151, −8.793673178662131154352783127561, −7.975424355213324118224597344090, −7.70180819667229364920721371508, −7.22605366021219881869706516704, −6.31317906131062753753098744230, −5.51312693814922998405834085291, −5.46386451791316632437277941259, −4.44315597227222548493466358119, −4.23444943724310348445779606696, −3.39606622370191707662506827238, −2.45212168631994686131440596548, −1.38666394890942631136185453236, −0.846002018288205718916269616300, 0.846002018288205718916269616300, 1.38666394890942631136185453236, 2.45212168631994686131440596548, 3.39606622370191707662506827238, 4.23444943724310348445779606696, 4.44315597227222548493466358119, 5.46386451791316632437277941259, 5.51312693814922998405834085291, 6.31317906131062753753098744230, 7.22605366021219881869706516704, 7.70180819667229364920721371508, 7.975424355213324118224597344090, 8.793673178662131154352783127561, 8.890664913287295193998741414151, 9.740547522904466300445649163495, 9.789727895342154361174443507442, 11.01651792981626866405587340870, 11.06625406840793981912470562700, 11.40357111009650033936710953608, 11.79500268335814591671195664500

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