Properties

Label 2-165-55.14-c1-0-0
Degree $2$
Conductor $165$
Sign $0.990 + 0.134i$
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  + (−1.60 − 2.20i)2-s + (−0.951 + 0.309i)3-s + (−1.67 + 5.16i)4-s + (−2.23 − 0.0757i)5-s + (2.20 + 1.60i)6-s + (0.462 + 0.150i)7-s + (8.89 − 2.88i)8-s + (0.809 − 0.587i)9-s + (3.41 + 5.04i)10-s + (3.06 + 1.26i)11-s − 5.43i·12-s + (2.12 + 2.92i)13-s + (−0.409 − 1.26i)14-s + (2.14 − 0.618i)15-s + (−11.8 − 8.59i)16-s + (−2.14 + 2.94i)17-s + ⋯
L(s)  = 1  + (−1.13 − 1.55i)2-s + (−0.549 + 0.178i)3-s + (−0.839 + 2.58i)4-s + (−0.999 − 0.0338i)5-s + (0.900 + 0.654i)6-s + (0.174 + 0.0568i)7-s + (3.14 − 1.02i)8-s + (0.269 − 0.195i)9-s + (1.07 + 1.59i)10-s + (0.924 + 0.381i)11-s − 1.56i·12-s + (0.588 + 0.810i)13-s + (−0.109 − 0.336i)14-s + (0.554 − 0.159i)15-s + (−2.95 − 2.14i)16-s + (−0.519 + 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.990 + 0.134i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.990 + 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.404546 - 0.0273474i\)
\(L(\frac12)\) \(\approx\) \(0.404546 - 0.0273474i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.0757i)T \)
11 \( 1 + (-3.06 - 1.26i)T \)
good2 \( 1 + (1.60 + 2.20i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + (-0.462 - 0.150i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.12 - 2.92i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.14 - 2.94i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.504 - 1.55i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.54iT - 23T^{2} \)
29 \( 1 + (0.326 - 1.00i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (4.93 - 3.58i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-8.17 - 2.65i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.39 - 10.4i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.41iT - 43T^{2} \)
47 \( 1 + (4.95 - 1.61i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.875 + 1.20i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.46 + 7.58i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-8.78 - 6.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.432iT - 67T^{2} \)
71 \( 1 + (4.86 + 3.53i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.10 - 0.359i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.57 - 2.60i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.86 - 2.57i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.38T + 89T^{2} \)
97 \( 1 + (-0.822 - 1.13i)T + (-29.9 + 92.2i)T^{2} \)
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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

−12.29487997685276252954406793904, −11.40415530514166219765935981873, −11.17644766649630831648667742558, −9.895897112441247042692360248133, −8.984661538839551309480688126325, −8.072398721793404315939753918594, −6.84723302400776284242163854752, −4.40536884678750651847589388498, −3.58013885831474209693525103940, −1.49976851255489888021833872316, 0.67260973755935056253684694863, 4.40721581348214180297179848456, 5.73392295349008015950369209737, 6.73833954258528005486956287558, 7.60928947789463730392070533224, 8.498331085559639484454120352128, 9.436235379879075018131457956781, 10.80134342770424330041671884649, 11.41995535178042422596901758184, 13.03129617314872819539789890228

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