Properties

Label 4-462e2-1.1-c3e2-0-3
Degree $4$
Conductor $213444$
Sign $1$
Analytic cond. $743.046$
Root an. cond. $5.22100$
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  + 4·2-s − 6·3-s + 12·4-s − 3·5-s − 24·6-s − 14·7-s + 32·8-s + 27·9-s − 12·10-s + 22·11-s − 72·12-s − 3·13-s − 56·14-s + 18·15-s + 80·16-s − 20·17-s + 108·18-s + 75·19-s − 36·20-s + 84·21-s + 88·22-s + 172·23-s − 192·24-s − 39·25-s − 12·26-s − 108·27-s − 168·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.268·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 0.379·10-s + 0.603·11-s − 1.73·12-s − 0.0640·13-s − 1.06·14-s + 0.309·15-s + 5/4·16-s − 0.285·17-s + 1.41·18-s + 0.905·19-s − 0.402·20-s + 0.872·21-s + 0.852·22-s + 1.55·23-s − 1.63·24-s − 0.311·25-s − 0.0905·26-s − 0.769·27-s − 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(743.046\)
Root analytic conductor: \(5.22100\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 213444,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.301214707\)
\(L(\frac12)\) \(\approx\) \(5.301214707\)
\(L(\frac{5}{2})\) 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)$
bad2$C_1$ \( ( 1 - p T )^{2} \)
3$C_1$ \( ( 1 + p T )^{2} \)
7$C_1$ \( ( 1 + p T )^{2} \)
11$C_1$ \( ( 1 - p T )^{2} \)
good5$D_{4}$ \( 1 + 3 T + 48 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 4192 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
17$C_2$ \( ( 1 + 10 T + p^{3} T^{2} )^{2} \)
19$D_{4}$ \( 1 - 75 T + 13286 T^{2} - 75 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 172 T + 28462 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 15 T + 14316 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 74 T + 53598 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 455 T + 151224 T^{2} - 455 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 374 T + 165458 T^{2} - 374 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 476 T + 186246 T^{2} - 476 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 301 T + 228458 T^{2} - 301 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 254 T + 18946 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 537 T + 466306 T^{2} - 537 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 336318 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 45 T + 600194 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 956 T + 679598 T^{2} - 956 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 473 T + 828860 T^{2} + 473 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 224 T - 308578 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 98 T + 851038 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 796 T + 1516054 T^{2} + 796 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 586 T + 1727370 T^{2} + 586 p^{3} T^{3} + p^{6} T^{4} \)
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

−10.93749998498494204676868754814, −10.85504538173225943297409312056, −9.999869671544826806860602623648, −9.736522067231453794213725580540, −9.182147288135336338117875363457, −8.692361228176262206949117159076, −7.63097186873670595593358645831, −7.56695666819891525861688570142, −6.88578522130028427368397636314, −6.60780470870861509995162722602, −5.90686793873481902482008844168, −5.82219206389564685166290295904, −5.13443489514564423209843254223, −4.64343217571334824198318220673, −3.99523803489251238832597774196, −3.82347558203724999627932901926, −2.77984079112836519625914137999, −2.50412614825025097398695565233, −1.13847112901828113289567030257, −0.74818393336867489295058753821, 0.74818393336867489295058753821, 1.13847112901828113289567030257, 2.50412614825025097398695565233, 2.77984079112836519625914137999, 3.82347558203724999627932901926, 3.99523803489251238832597774196, 4.64343217571334824198318220673, 5.13443489514564423209843254223, 5.82219206389564685166290295904, 5.90686793873481902482008844168, 6.60780470870861509995162722602, 6.88578522130028427368397636314, 7.56695666819891525861688570142, 7.63097186873670595593358645831, 8.692361228176262206949117159076, 9.182147288135336338117875363457, 9.736522067231453794213725580540, 9.999869671544826806860602623648, 10.85504538173225943297409312056, 10.93749998498494204676868754814

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