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Properties

Label 2-165-165.29-c1-0-5
Degree $2$
Conductor $165$
Sign $0.737 - 0.675i$
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  + (−0.413 + 0.568i)2-s + (−1.53 − 0.807i)3-s + (0.465 + 1.43i)4-s + (1.55 − 1.60i)5-s + (1.09 − 0.538i)6-s + (0.615 + 1.89i)7-s + (−2.34 − 0.761i)8-s + (1.69 + 2.47i)9-s + (0.269 + 1.54i)10-s + (3.26 − 0.597i)11-s + (0.442 − 2.56i)12-s + (3.43 + 2.49i)13-s + (−1.33 − 0.432i)14-s + (−3.68 + 1.20i)15-s + (−1.03 + 0.750i)16-s + (2.51 + 3.45i)17-s + ⋯
L(s)  = 1  + (−0.292 + 0.402i)2-s + (−0.884 − 0.465i)3-s + (0.232 + 0.715i)4-s + (0.696 − 0.717i)5-s + (0.446 − 0.219i)6-s + (0.232 + 0.715i)7-s + (−0.828 − 0.269i)8-s + (0.565 + 0.824i)9-s + (0.0850 + 0.489i)10-s + (0.983 − 0.180i)11-s + (0.127 − 0.741i)12-s + (0.952 + 0.691i)13-s + (−0.355 − 0.115i)14-s + (−0.950 + 0.310i)15-s + (−0.258 + 0.187i)16-s + (0.609 + 0.838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.737 - 0.675i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.737 - 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870082 + 0.338098i\)
\(L(\frac12)\) \(\approx\) \(0.870082 + 0.338098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 + 0.807i)T \)
5 \( 1 + (-1.55 + 1.60i)T \)
11 \( 1 + (-3.26 + 0.597i)T \)
good2 \( 1 + (0.413 - 0.568i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.615 - 1.89i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.43 - 2.49i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.51 - 3.45i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.257 - 0.0836i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.30T + 23T^{2} \)
29 \( 1 + (2.26 + 6.96i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.24 + 3.81i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-5.66 + 1.84i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.95 - 9.08i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.68T + 43T^{2} \)
47 \( 1 + (0.0723 - 0.222i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.61 + 4.07i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (10.0 - 3.26i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.68 - 7.81i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.901iT - 67T^{2} \)
71 \( 1 + (-0.309 - 0.425i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.18 + 9.79i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.132 + 0.182i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.18 - 1.62i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 15.8iT - 89T^{2} \)
97 \( 1 + (1.94 - 2.67i)T + (-29.9 - 92.2i)T^{2} \)
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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

−12.79548647553237173698468005416, −11.93143192457325055301876442192, −11.35034784607666269305350471541, −9.716204487882029593157338101909, −8.714984792128482543291463903843, −7.81422026060023149353379035979, −6.27666184147190042277429444902, −5.91375287362474935428661109919, −4.14355080741747584350457403458, −1.79141891035734549825632966466, 1.34023159427640740591517465856, 3.52989786165288663267710579165, 5.27804298731629151777192914552, 6.19730052229038380637415026350, 7.13351845412140671923937296406, 9.160918872201945460060136150829, 9.976528262496899736202552340532, 10.75233293835904097086898615739, 11.27158131486145703609678867722, 12.40592522497698062810827928768

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