Properties

Degree $2$
Conductor $1110$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 4·11-s − 12-s − 2·13-s − 15-s + 16-s + 2·17-s + 18-s + 4·19-s + 20-s + 4·22-s − 8·23-s − 24-s + 25-s − 2·26-s − 27-s − 2·29-s − 30-s + 8·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.371·29-s − 0.182·30-s + 1.43·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{1110} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.442342719\)
\(L(\frac12)\) \(\approx\) \(2.442342719\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
37 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.94545701039644, −19.45027408555040, −18.55889376275282, −17.60989293026216, −17.32491439020037, −16.32087667619005, −15.98715696698484, −14.92114942397498, −14.21282771927731, −13.82600098729984, −12.79525044017814, −12.04324212692577, −11.72335954133093, −10.73188641272033, −9.860059891719032, −9.312841926862842, −7.964758105018484, −7.163354274888833, −6.174017211556287, −5.741565906440673, −4.624212162495126, −3.852823030337765, −2.544724752943126, −1.223276202553147, 1.223276202553147, 2.544724752943126, 3.852823030337765, 4.624212162495126, 5.741565906440673, 6.174017211556287, 7.163354274888833, 7.964758105018484, 9.312841926862842, 9.860059891719032, 10.73188641272033, 11.72335954133093, 12.04324212692577, 12.79525044017814, 13.82600098729984, 14.21282771927731, 14.92114942397498, 15.98715696698484, 16.32087667619005, 17.32491439020037, 17.60989293026216, 18.55889376275282, 19.45027408555040, 19.94545701039644

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