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Properties

Label 2-165-55.14-c1-0-9
Degree $2$
Conductor $165$
Sign $-0.971 + 0.236i$
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  + (−1.38 − 1.91i)2-s + (0.951 − 0.309i)3-s + (−1.10 + 3.41i)4-s + (0.389 − 2.20i)5-s + (−1.91 − 1.38i)6-s + (−1.63 − 0.529i)7-s + (3.56 − 1.15i)8-s + (0.809 − 0.587i)9-s + (−4.75 + 2.31i)10-s + (−0.406 − 3.29i)11-s + 3.58i·12-s + (−2.07 − 2.85i)13-s + (1.25 + 3.85i)14-s + (−0.310 − 2.21i)15-s + (−1.37 − 0.996i)16-s + (−0.219 + 0.302i)17-s + ⋯
L(s)  = 1  + (−0.982 − 1.35i)2-s + (0.549 − 0.178i)3-s + (−0.554 + 1.70i)4-s + (0.174 − 0.984i)5-s + (−0.780 − 0.567i)6-s + (−0.616 − 0.200i)7-s + (1.26 − 0.409i)8-s + (0.269 − 0.195i)9-s + (−1.50 + 0.731i)10-s + (−0.122 − 0.992i)11-s + 1.03i·12-s + (−0.576 − 0.792i)13-s + (0.334 + 1.02i)14-s + (−0.0800 − 0.571i)15-s + (−0.342 − 0.249i)16-s + (−0.0532 + 0.0732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.971 + 0.236i$
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: \((2,\ 165,\ (\ :1/2),\ -0.971 + 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0856805 - 0.715819i\)
\(L(\frac12)\) \(\approx\) \(0.0856805 - 0.715819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.389 + 2.20i)T \)
11 \( 1 + (0.406 + 3.29i)T \)
good2 \( 1 + (1.38 + 1.91i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + (1.63 + 0.529i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.07 + 2.85i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.219 - 0.302i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.78 - 5.48i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.80iT - 23T^{2} \)
29 \( 1 + (-1.08 + 3.34i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.16 + 5.93i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-10.9 - 3.56i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.747 + 2.30i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.224iT - 43T^{2} \)
47 \( 1 + (-4.44 + 1.44i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.43 + 1.97i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.65 - 5.09i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.75 - 2.00i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 9.08iT - 67T^{2} \)
71 \( 1 + (-7.09 - 5.15i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-14.2 - 4.62i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.11 + 5.16i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.37 - 6.02i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (8.61 + 11.8i)T + (-29.9 + 92.2i)T^{2} \)
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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

−12.27453149480193163185609089224, −11.38165806312787462745504597709, −9.923086383625014785323894384366, −9.707131782885318968098553620463, −8.391216555920038587174730146899, −7.898506925299156863722345873659, −5.83320637508115426476812629930, −3.89379396954641769056920237191, −2.66134810410926806241912196655, −0.920723311772387591494465935399, 2.68749052746519724636558182875, 4.79159892670375116130303307349, 6.48972527029764299516251215051, 6.97554180550036215552831826585, 7.999532132960216220012699739562, 9.353519392635018679345202596203, 9.680719141704913827908827900883, 10.81626111148179725970386760631, 12.39063365930067794523260892333, 13.82085459650761788126019869900

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