Properties

Label 4-546e2-1.1-c1e2-0-39
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  − 2·3-s − 4-s + 3·9-s + 2·12-s + 6·13-s + 16-s + 2·17-s + 6·23-s + 9·25-s − 4·27-s + 18·29-s − 3·36-s − 12·39-s + 14·43-s − 2·48-s − 49-s − 4·51-s − 6·52-s − 20·53-s + 22·61-s − 64-s − 2·68-s − 12·69-s − 18·75-s − 24·79-s + 5·81-s − 36·87-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 9-s + 0.577·12-s + 1.66·13-s + 1/4·16-s + 0.485·17-s + 1.25·23-s + 9/5·25-s − 0.769·27-s + 3.34·29-s − 1/2·36-s − 1.92·39-s + 2.13·43-s − 0.288·48-s − 1/7·49-s − 0.560·51-s − 0.832·52-s − 2.74·53-s + 2.81·61-s − 1/8·64-s − 0.242·68-s − 1.44·69-s − 2.07·75-s − 2.70·79-s + 5/9·81-s − 3.85·87-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

\(L(1)\) \(\approx\) \(1.409753685\)
\(L(\frac12)\) \(\approx\) \(1.409753685\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good5$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−11.04505362659265251242170856084, −10.78104301170401505307511686094, −10.03956400679909049827172074353, −10.02004710064398676615273167938, −9.219480901588857273058422341604, −8.783531309560911449799472382690, −8.383785026984063472299012064678, −8.109639533376156155836635696537, −7.13626991888380856938724060512, −6.98106462795789150955807148679, −6.23283583615769451213451930779, −6.16423625528239770109262574331, −5.42033514195581625818776081159, −4.89211644729518278801671489593, −4.58595412521107281643708288977, −3.98271519982292327097434245597, −3.18728483142858966879703378545, −2.72104849752177574724365641659, −1.11881376874371552172747247000, −1.03407213013071041944117832355, 1.03407213013071041944117832355, 1.11881376874371552172747247000, 2.72104849752177574724365641659, 3.18728483142858966879703378545, 3.98271519982292327097434245597, 4.58595412521107281643708288977, 4.89211644729518278801671489593, 5.42033514195581625818776081159, 6.16423625528239770109262574331, 6.23283583615769451213451930779, 6.98106462795789150955807148679, 7.13626991888380856938724060512, 8.109639533376156155836635696537, 8.383785026984063472299012064678, 8.783531309560911449799472382690, 9.219480901588857273058422341604, 10.02004710064398676615273167938, 10.03956400679909049827172074353, 10.78104301170401505307511686094, 11.04505362659265251242170856084

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