↑

Properties

Label 2-11e2-11.10-c2-0-9
Degree $2$
Conductor $121$
Sign $-0.522 + 0.852i$
Analytic cond. $3.29701$
Root an. cond. $1.81576$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.16i·2-s − 2.85·3-s − 0.674·4-s + 7.64·5-s + 6.16i·6-s − 6.05i·7-s − 7.18i·8-s − 0.870·9-s − 16.5i·10-s + 1.92·12-s + 3.21i·13-s − 13.0·14-s − 21.7·15-s − 18.2·16-s − 6.22i·17-s + 1.88i·18-s + ⋯
L(s)  = 1  − 1.08i·2-s − 0.950·3-s − 0.168·4-s + 1.52·5-s + 1.02i·6-s − 0.865i·7-s − 0.898i·8-s − 0.0967·9-s − 1.65i·10-s + 0.160·12-s + 0.247i·13-s − 0.935·14-s − 1.45·15-s − 1.14·16-s − 0.366i·17-s + 0.104i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(3.29701\)
Root analytic conductor: \(1.81576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (120, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1),\ -0.522 + 0.852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.645371 - 1.15197i\)
\(L(\frac12)\) \(\approx\) \(0.645371 - 1.15197i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + 2.16iT - 4T^{2} \)
3 \( 1 + 2.85T + 9T^{2} \)
5 \( 1 - 7.64T + 25T^{2} \)
7 \( 1 + 6.05iT - 49T^{2} \)
13 \( 1 - 3.21iT - 169T^{2} \)
17 \( 1 + 6.22iT - 289T^{2} \)
19 \( 1 + 19.4iT - 361T^{2} \)
23 \( 1 - 3.47T + 529T^{2} \)
29 \( 1 - 36.1iT - 841T^{2} \)
31 \( 1 + 3.28T + 961T^{2} \)
37 \( 1 - 63.2T + 1.36e3T^{2} \)
41 \( 1 - 31.9iT - 1.68e3T^{2} \)
43 \( 1 - 43.9iT - 1.84e3T^{2} \)
47 \( 1 + 34.7T + 2.20e3T^{2} \)
53 \( 1 - 16.5T + 2.80e3T^{2} \)
59 \( 1 + 63.7T + 3.48e3T^{2} \)
61 \( 1 - 110. iT - 3.72e3T^{2} \)
67 \( 1 - 96.1T + 4.48e3T^{2} \)
71 \( 1 + 44.8T + 5.04e3T^{2} \)
73 \( 1 - 63.2iT - 5.32e3T^{2} \)
79 \( 1 + 14.1iT - 6.24e3T^{2} \)
83 \( 1 + 88.6iT - 6.88e3T^{2} \)
89 \( 1 - 51.3T + 7.92e3T^{2} \)
97 \( 1 - 31.5T + 9.40e3T^{2} \)
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

−12.84256912766722500662682639174, −11.55516918093018197508264160392, −10.88580531058031722084465396774, −10.07676590827454142494191600771, −9.188808486044041479012583245459, −6.99889492045472030304536924213, −6.08225049326900337058202181901, −4.72717042117219729735756471226, −2.76131401075080902000699762044, −1.12216600829100047810058096698, 2.24944061965974318447714906744, 5.21314104837995119814123622545, 5.90371670674491066837617749963, 6.40119611668876416926238533546, 8.052171208467582552165193757506, 9.244807116593390624459905678703, 10.41794603745099880241173467051, 11.52059397903460133800555140038, 12.57480972794996357393924047393, 13.80587540142108365226590144154

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