Properties

Label 2-165-55.4-c1-0-9
Degree $2$
Conductor $165$
Sign $0.302 + 0.953i$
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  + (1.12 − 1.54i)2-s + (0.951 + 0.309i)3-s + (−0.506 − 1.55i)4-s + (1.25 − 1.85i)5-s + (1.54 − 1.12i)6-s + (−4.23 + 1.37i)7-s + (0.656 + 0.213i)8-s + (0.809 + 0.587i)9-s + (−1.44 − 4.01i)10-s + (−1.22 + 3.08i)11-s − 1.63i·12-s + (0.313 − 0.432i)13-s + (−2.62 + 8.08i)14-s + (1.76 − 1.37i)15-s + (3.71 − 2.69i)16-s + (−2.67 − 3.67i)17-s + ⋯
L(s)  = 1  + (0.792 − 1.09i)2-s + (0.549 + 0.178i)3-s + (−0.253 − 0.779i)4-s + (0.561 − 0.827i)5-s + (0.629 − 0.457i)6-s + (−1.60 + 0.520i)7-s + (0.231 + 0.0753i)8-s + (0.269 + 0.195i)9-s + (−0.458 − 1.26i)10-s + (−0.368 + 0.929i)11-s − 0.472i·12-s + (0.0870 − 0.119i)13-s + (−0.702 + 2.16i)14-s + (0.455 − 0.354i)15-s + (0.928 − 0.674i)16-s + (−0.647 − 0.891i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.302 + 0.953i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.302 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50759 - 1.10312i\)
\(L(\frac12)\) \(\approx\) \(1.50759 - 1.10312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (-1.25 + 1.85i)T \)
11 \( 1 + (1.22 - 3.08i)T \)
good2 \( 1 + (-1.12 + 1.54i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (4.23 - 1.37i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.313 + 0.432i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.67 + 3.67i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.594 - 1.83i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.18iT - 23T^{2} \)
29 \( 1 + (-1.46 - 4.52i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.82 + 2.05i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.501 + 0.162i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.45 + 4.46i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.02iT - 43T^{2} \)
47 \( 1 + (-11.4 - 3.70i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.95 - 4.06i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.21 + 12.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (9.45 - 6.87i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 11.5iT - 67T^{2} \)
71 \( 1 + (2.72 - 1.97i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.618 - 0.200i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (14.3 + 10.4i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.49 - 3.43i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.24T + 89T^{2} \)
97 \( 1 + (-0.669 + 0.920i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.66508074497194743477142510455, −12.10863589649723038535662392837, −10.58358357056942072602304115195, −9.643818520443324178363247946573, −9.075308414961991279966396437767, −7.38606949648871069074782121291, −5.76718457696977187115954923633, −4.59323983571603853917798043155, −3.27540933059297955337404829608, −2.13393416303128332419964537225, 2.89237772685172649075156724519, 4.07639994851599338916435955437, 5.99045482407366365310925062048, 6.48072692857079486865912181781, 7.38432472545126440639299723997, 8.748760536706121238226378046979, 10.05850620662517289596149789527, 10.79590197649693268301556140986, 12.74626318474654805414833622454, 13.37655563442462166171439320101

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