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Properties

Label 2-1320-11.4-c1-0-8
Degree $2$
Conductor $1320$
Sign $0.970 - 0.242i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)3-s + (0.809 − 0.587i)5-s + (0.236 + 0.726i)7-s + (−0.809 − 0.587i)9-s + (3.04 + 1.31i)11-s + (4.73 + 3.44i)13-s + (−0.309 − 0.951i)15-s + (−6.16 + 4.47i)17-s + (−2.38 + 7.33i)19-s + 0.763·21-s + 7.61·23-s + (0.309 − 0.951i)25-s + (−0.809 + 0.587i)27-s + (−0.809 − 2.48i)29-s + (1.11 + 0.812i)31-s + ⋯
L(s)  = 1  + (0.178 − 0.549i)3-s + (0.361 − 0.262i)5-s + (0.0892 + 0.274i)7-s + (−0.269 − 0.195i)9-s + (0.918 + 0.396i)11-s + (1.31 + 0.954i)13-s + (−0.0797 − 0.245i)15-s + (−1.49 + 1.08i)17-s + (−0.546 + 1.68i)19-s + 0.166·21-s + 1.58·23-s + (0.0618 − 0.190i)25-s + (−0.155 + 0.113i)27-s + (−0.150 − 0.462i)29-s + (0.200 + 0.145i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.970 - 0.242i$
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: \((2,\ 1320,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.994446314\)
\(L(\frac12)\) \(\approx\) \(1.994446314\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.04 - 1.31i)T \)
good7 \( 1 + (-0.236 - 0.726i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.73 - 3.44i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.16 - 4.47i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.38 - 7.33i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.61T + 23T^{2} \)
29 \( 1 + (0.809 + 2.48i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.11 - 0.812i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.972 + 2.99i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.854 + 2.62i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.32T + 43T^{2} \)
47 \( 1 + (-0.572 + 1.76i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.61 - 1.17i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.354 - 1.08i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (10.4 - 7.60i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 15.5T + 67T^{2} \)
71 \( 1 + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.14 + 6.60i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.78 - 4.20i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.47 + 4.70i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.23T + 89T^{2} \)
97 \( 1 + (9.09 + 6.60i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.390271085412432312460156755001, −8.763251849656354018566742316092, −8.332378963019535327018655792956, −7.00974946816908279233160614095, −6.41124158539066140422942580259, −5.75467499838648262005227939669, −4.36898806183121812968621525665, −3.69389939103109713504884390591, −2.08216086008798193308182354172, −1.42925092985306223727873298812, 0.907268419458868295573186266966, 2.58461704059945877190505257873, 3.41671770822027582616929991666, 4.50239760415966084056603939491, 5.26652005991497531524159193121, 6.49858090769500093112824254072, 6.88203164629428727461525623226, 8.194119159991469157814075266723, 9.108485065154399847544996347908, 9.236956770503022320061335512593

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