Properties

Label 2-1122-1.1-c1-0-18
Degree $2$
Conductor $1122$
Sign $-1$
Analytic cond. $8.95921$
Root an. cond. $2.99319$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 2·5-s − 6-s − 2·7-s − 8-s + 9-s + 2·10-s + 11-s + 12-s + 2·14-s − 2·15-s + 16-s − 17-s − 18-s + 6·19-s − 2·20-s − 2·21-s − 22-s − 6·23-s − 24-s − 25-s + 27-s − 2·28-s − 2·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s + 0.534·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.37·19-s − 0.447·20-s − 0.436·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $-1$
Analytic conductor: \(8.95921\)
Root analytic conductor: \(2.99319\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−9.660611426738837831990429507656, −8.411437040935556632941293755633, −8.022550439402926346291352599484, −7.06302018923147745948865165948, −6.41012704790367578981180953374, −5.07613970596192180401858163476, −3.72750128626488492549930290997, −3.18681322176319844452732593618, −1.72438918850306340042530588235, 0, 1.72438918850306340042530588235, 3.18681322176319844452732593618, 3.72750128626488492549930290997, 5.07613970596192180401858163476, 6.41012704790367578981180953374, 7.06302018923147745948865165948, 8.022550439402926346291352599484, 8.411437040935556632941293755633, 9.660611426738837831990429507656

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