↑

Properties

Label 2-1320-11.3-c1-0-16
Degree $2$
Conductor $1320$
Sign $0.693 + 0.720i$
Analytic cond. $10.5402$
Root an. cond. $3.24657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)5-s + (0.354 − 1.08i)7-s + (−0.809 + 0.587i)9-s + (3.27 + 0.501i)11-s + (2.21 − 1.60i)13-s + (0.309 − 0.951i)15-s + (−3.86 − 2.80i)17-s + (1.47 + 4.52i)19-s − 1.14·21-s + 5.11·23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (1.17 − 3.60i)29-s + (−5.04 + 3.66i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s + (0.361 + 0.262i)5-s + (0.133 − 0.411i)7-s + (−0.269 + 0.195i)9-s + (0.988 + 0.151i)11-s + (0.613 − 0.445i)13-s + (0.0797 − 0.245i)15-s + (−0.937 − 0.681i)17-s + (0.337 + 1.03i)19-s − 0.249·21-s + 1.06·23-s + (0.0618 + 0.190i)25-s + (0.155 + 0.113i)27-s + (0.217 − 0.669i)29-s + (−0.905 + 0.657i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.693 + 0.720i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.693 + 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.802995798\)
\(L(\frac12)\) \(\approx\) \(1.802995798\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-3.27 - 0.501i)T \)
good7 \( 1 + (-0.354 + 1.08i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.21 + 1.60i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.86 + 2.80i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.47 - 4.52i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.11T + 23T^{2} \)
29 \( 1 + (-1.17 + 3.60i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.04 - 3.66i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.96 + 6.03i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.48 + 4.55i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + (-0.994 - 3.05i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.94 + 2.13i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.55 + 4.79i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.42 + 1.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.11T + 67T^{2} \)
71 \( 1 + (1.29 + 0.940i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.88 + 5.81i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-13.2 + 9.64i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.88 + 4.99i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 0.628T + 89T^{2} \)
97 \( 1 + (-6.28 + 4.56i)T + (29.9 - 92.2i)T^{2} \)
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

−9.349414503783073856883552664453, −8.870334349546808494287474912008, −7.69498586541646511891128639570, −7.09403974752379040915631758772, −6.27166192419356678590286500972, −5.52922536602511157353267273061, −4.35802577220273301641167807413, −3.36031299244041625684277519308, −2.09264851662722751206743534831, −0.937959208240632817776638224354, 1.21707809890791194092309524971, 2.57053257493054511386216689411, 3.81811491044065406055605625535, 4.60750170032426972866706315523, 5.54185069332270783083963287044, 6.37563333390881656416126941224, 7.09751586532765894982912439970, 8.507682608800158626988394093813, 9.007657958989479663175488363563, 9.485707507164042049735056941403

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