Properties

Label 4-546e2-1.1-c1e2-0-62
Degree $4$
Conductor $298116$
Sign $1$
Analytic cond. $19.0081$
Root an. cond. $2.08802$
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  + 2-s + 3-s + 8·5-s + 6-s − 7-s − 8-s + 8·10-s + 3·11-s − 5·13-s − 14-s + 8·15-s − 16-s + 5·17-s − 3·19-s − 21-s + 3·22-s − 6·23-s − 24-s + 38·25-s − 5·26-s − 27-s + 9·29-s + 8·30-s − 8·31-s + 3·33-s + 5·34-s − 8·35-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 3.57·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 2.52·10-s + 0.904·11-s − 1.38·13-s − 0.267·14-s + 2.06·15-s − 1/4·16-s + 1.21·17-s − 0.688·19-s − 0.218·21-s + 0.639·22-s − 1.25·23-s − 0.204·24-s + 38/5·25-s − 0.980·26-s − 0.192·27-s + 1.67·29-s + 1.46·30-s − 1.43·31-s + 0.522·33-s + 0.857·34-s − 1.35·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(298116\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.0081\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{546} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.528997866\)
\(L(\frac12)\) \(\approx\) \(5.528997866\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
13$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 12 T + 47 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−10.65234761240300713044289414016, −10.48036719555506605811388077778, −9.909368445830808424276765756360, −9.702530217671609123200659693485, −9.387213486985284318908274632670, −9.182987224855806064294159330136, −8.503923422612367660369418439499, −8.013122467384839701530357397457, −7.23088041632629343234814940258, −6.51905856959987526073644511782, −6.32663603249249366271002008719, −6.12047470895864318007447049586, −5.40245095979644550487900414957, −4.98948553865467236865492509446, −4.77580161198113373617957070828, −3.56419567736335344125202481975, −3.15942315338852056502104693645, −2.32812910948998003646911401521, −2.06651423055000875473771703220, −1.41657840200847134643381722163, 1.41657840200847134643381722163, 2.06651423055000875473771703220, 2.32812910948998003646911401521, 3.15942315338852056502104693645, 3.56419567736335344125202481975, 4.77580161198113373617957070828, 4.98948553865467236865492509446, 5.40245095979644550487900414957, 6.12047470895864318007447049586, 6.32663603249249366271002008719, 6.51905856959987526073644511782, 7.23088041632629343234814940258, 8.013122467384839701530357397457, 8.503923422612367660369418439499, 9.182987224855806064294159330136, 9.387213486985284318908274632670, 9.702530217671609123200659693485, 9.909368445830808424276765756360, 10.48036719555506605811388077778, 10.65234761240300713044289414016

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