Properties

Label 4-546e2-1.1-c1e2-0-27
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 + 2·5-s + 6-s − 7-s − 8-s + 2·10-s − 3·11-s − 2·13-s − 14-s + 2·15-s − 16-s − 17-s + 3·19-s − 21-s − 3·22-s − 24-s + 5·25-s − 2·26-s − 27-s + 6·29-s + 2·30-s + 4·31-s − 3·33-s − 34-s − 2·35-s + 2·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.632·10-s − 0.904·11-s − 0.554·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s − 0.242·17-s + 0.688·19-s − 0.218·21-s − 0.639·22-s − 0.204·24-s + 25-s − 0.392·26-s − 0.192·27-s + 1.11·29-s + 0.365·30-s + 0.718·31-s − 0.522·33-s − 0.171·34-s − 0.338·35-s + 0.328·37-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 298116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.124943938\)
\(L(\frac12)\) \(\approx\) \(3.124943938\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−11.11627975423796464173872017574, −10.58948785133078455353567380320, −9.992170014811414547373982348184, −9.731708604252730298311664446684, −9.253347824562948930149468558988, −9.039277001127544061606992002705, −8.160893328886625656840873248713, −8.084624562723368102394740189153, −7.41041319569221088915125316641, −6.85657401507790752482144449679, −6.41941987325533303280365592014, −5.81242054577302468369073997872, −5.52874986029181995125765103422, −4.78112701832998446623982469398, −4.60167290071852992059786229672, −3.75994407612231324242106755751, −3.05306367969322563837806220540, −2.63931602917772875355829287061, −2.20043645908680964663657032609, −0.911323824910058518842134785462, 0.911323824910058518842134785462, 2.20043645908680964663657032609, 2.63931602917772875355829287061, 3.05306367969322563837806220540, 3.75994407612231324242106755751, 4.60167290071852992059786229672, 4.78112701832998446623982469398, 5.52874986029181995125765103422, 5.81242054577302468369073997872, 6.41941987325533303280365592014, 6.85657401507790752482144449679, 7.41041319569221088915125316641, 8.084624562723368102394740189153, 8.160893328886625656840873248713, 9.039277001127544061606992002705, 9.253347824562948930149468558988, 9.731708604252730298311664446684, 9.992170014811414547373982348184, 10.58948785133078455353567380320, 11.11627975423796464173872017574

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