↑

Properties

Label 2-1320-11.4-c1-0-12
Degree $2$
Conductor $1320$
Sign $0.852 + 0.523i$
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 + (0.646 + 1.99i)7-s + (−0.809 − 0.587i)9-s + (1.19 + 3.09i)11-s + (−2.47 − 1.79i)13-s + (−0.309 − 0.951i)15-s + (3.88 − 2.82i)17-s + (1.97 − 6.07i)19-s + 2.09·21-s + 1.36·23-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (0.438 + 1.34i)29-s + (6.59 + 4.79i)31-s + ⋯
L(s)  = 1  + (0.178 − 0.549i)3-s + (0.361 − 0.262i)5-s + (0.244 + 0.752i)7-s + (−0.269 − 0.195i)9-s + (0.359 + 0.933i)11-s + (−0.686 − 0.498i)13-s + (−0.0797 − 0.245i)15-s + (0.942 − 0.684i)17-s + (0.452 − 1.39i)19-s + 0.456·21-s + 0.284·23-s + (0.0618 − 0.190i)25-s + (−0.155 + 0.113i)27-s + (0.0814 + 0.250i)29-s + (1.18 + 0.860i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.852 + 0.523i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.852 + 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.004479537\)
\(L(\frac12)\) \(\approx\) \(2.004479537\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.19 - 3.09i)T \)
good7 \( 1 + (-0.646 - 1.99i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.47 + 1.79i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.88 + 2.82i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.97 + 6.07i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.36T + 23T^{2} \)
29 \( 1 + (-0.438 - 1.34i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.59 - 4.79i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.66 - 8.20i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.53 - 4.72i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + (-3.34 + 10.3i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.90 + 7.19i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.21 + 3.74i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.62 - 1.17i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 8.98T + 67T^{2} \)
71 \( 1 + (-9.17 + 6.66i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.983 - 3.02i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.19 - 1.59i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.36 + 4.62i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + (-2.23 - 1.62i)T + (29.9 + 92.2i)T^{2} \)
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

−9.505020066635964047654920861591, −8.806996648874742395665605195829, −7.896596301961525004744263395054, −7.14674474753821971467350853453, −6.36391564074432492749232842492, −5.19270139018804579800897256562, −4.78797324691342427132131832905, −3.08463390284181898165824472937, −2.33058180413418838423264800680, −1.03577811543762695028744296764, 1.17248847699306448437302517943, 2.63442472696380243185049640914, 3.73791667019980944805366860867, 4.37674195938251968798145656316, 5.67935320102930615272787564439, 6.17136802665222477994485907692, 7.51849489021617816180240367430, 7.930335198943782720692801382064, 9.107481106411027910690049010027, 9.682984120906582705563809537625

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