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Properties

Label 2-1320-11.3-c1-0-18
Degree $2$
Conductor $1320$
Sign $0.585 + 0.810i$
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 + (1.18 − 3.65i)7-s + (−0.809 + 0.587i)9-s + (3.17 + 0.958i)11-s + (−1.87 + 1.36i)13-s + (0.309 − 0.951i)15-s + (0.113 + 0.0823i)17-s + (−1.27 − 3.92i)19-s + 3.83·21-s − 1.31·23-s + (0.309 + 0.951i)25-s + (−0.809 − 0.587i)27-s + (2.18 − 6.71i)29-s + (2.86 − 2.08i)31-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s + (−0.361 − 0.262i)5-s + (0.448 − 1.38i)7-s + (−0.269 + 0.195i)9-s + (0.957 + 0.288i)11-s + (−0.520 + 0.378i)13-s + (0.0797 − 0.245i)15-s + (0.0275 + 0.0199i)17-s + (−0.292 − 0.900i)19-s + 0.837·21-s − 0.275·23-s + (0.0618 + 0.190i)25-s + (−0.155 − 0.113i)27-s + (0.404 − 1.24i)29-s + (0.514 − 0.374i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.585 + 0.810i$
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: \((2,\ 1320,\ (\ :1/2),\ 0.585 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.591172984\)
\(L(\frac12)\) \(\approx\) \(1.591172984\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.17 - 0.958i)T \)
good7 \( 1 + (-1.18 + 3.65i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.87 - 1.36i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.113 - 0.0823i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.27 + 3.92i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.31T + 23T^{2} \)
29 \( 1 + (-2.18 + 6.71i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.86 + 2.08i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.82 + 8.68i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.13 + 3.48i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.24T + 43T^{2} \)
47 \( 1 + (-2.53 - 7.81i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.39 + 6.82i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.21 + 6.81i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.76 + 4.18i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.267T + 67T^{2} \)
71 \( 1 + (-10.4 - 7.62i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.24 + 3.81i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.73 - 1.25i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.76 + 3.46i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (7.65 - 5.56i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.587016185748419671302219055882, −8.763854202758119045716998780178, −7.85501210544580728513519314987, −7.17695414632621007175357195472, −6.32774180107841862473565889984, −4.94571896436494938408978615172, −4.27697649555998543545970732717, −3.74132346947917638556880708805, −2.20851412270202108945473462460, −0.69387620377079089012877486049, 1.41107656025259365307171802619, 2.55090200320399676692801311885, 3.46833733286693830406136361176, 4.74272547654389781209660446978, 5.71069460790139299268906366979, 6.45821173472012300518460508469, 7.31939256756985433582176020998, 8.439545602976325780949324712348, 8.554255503635325297084219120686, 9.666736724883632135230321477725

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