Properties

Label 2-165-11.4-c1-0-7
Degree $2$
Conductor $165$
Sign $0.171 + 0.985i$
Analytic cond. $1.31753$
Root an. cond. $1.14783$
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  + (0.212 + 0.154i)2-s + (0.309 − 0.951i)3-s + (−0.596 − 1.83i)4-s + (−0.809 + 0.587i)5-s + (0.212 − 0.154i)6-s + (−0.986 − 3.03i)7-s + (0.318 − 0.980i)8-s + (−0.809 − 0.587i)9-s − 0.262·10-s + (3.27 + 0.547i)11-s − 1.93·12-s + (0.905 + 0.658i)13-s + (0.258 − 0.796i)14-s + (0.309 + 0.951i)15-s + (−2.90 + 2.11i)16-s + (0.0713 − 0.0518i)17-s + ⋯
L(s)  = 1  + (0.150 + 0.109i)2-s + (0.178 − 0.549i)3-s + (−0.298 − 0.918i)4-s + (−0.361 + 0.262i)5-s + (0.0866 − 0.0629i)6-s + (−0.372 − 1.14i)7-s + (0.112 − 0.346i)8-s + (−0.269 − 0.195i)9-s − 0.0829·10-s + (0.986 + 0.165i)11-s − 0.557·12-s + (0.251 + 0.182i)13-s + (0.0691 − 0.212i)14-s + (0.0797 + 0.245i)15-s + (−0.726 + 0.527i)16-s + (0.0173 − 0.0125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.171 + 0.985i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.171 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.864477 - 0.727139i\)
\(L(\frac12)\) \(\approx\) \(0.864477 - 0.727139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-3.27 - 0.547i)T \)
good2 \( 1 + (-0.212 - 0.154i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.986 + 3.03i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.905 - 0.658i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.0713 + 0.0518i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0212 - 0.0654i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6.65T + 23T^{2} \)
29 \( 1 + (-1.15 - 3.55i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.75 - 5.63i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.57 + 7.92i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.60 - 11.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + (-0.280 + 0.863i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.705 + 0.512i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.567 - 1.74i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.13 + 5.91i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.53T + 67T^{2} \)
71 \( 1 + (-3.77 + 2.74i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.21 + 6.80i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.640 + 0.465i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.200 - 0.145i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + (-3.31 - 2.40i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.84489595786361414394325213729, −11.55412100494819050973282074452, −10.60000870435391085677402624547, −9.642252811206026008797556332478, −8.510737930208527260450712613369, −6.95340355655862809991183390431, −6.57428820747595171454761509043, −4.84477144858693161554434443541, −3.54102272851467987681702733628, −1.17355736768305559697348692290, 2.85066806787227402372414792078, 3.97060061287208196767713201781, 5.21426843447885058762701864968, 6.71613582051078145841064181078, 8.309579394142411968198592844448, 8.813395091118742318985703315996, 9.813670954184800872924567754082, 11.45787247163773312104381928556, 11.95566994054465433375893217595, 12.96566371282347067121047199073

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