Properties

Label 2-165-33.8-c1-0-11
Degree $2$
Conductor $165$
Sign $0.972 + 0.232i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.860 − 0.625i)2-s + (1.72 + 0.197i)3-s + (−0.268 + 0.825i)4-s + (0.587 − 0.809i)5-s + (1.60 − 0.905i)6-s + (−2.18 − 0.709i)7-s + (0.942 + 2.90i)8-s + (2.92 + 0.681i)9-s − 1.06i·10-s + (−2.26 − 2.41i)11-s + (−0.625 + 1.36i)12-s + (−1.13 − 1.55i)13-s + (−2.32 + 0.754i)14-s + (1.17 − 1.27i)15-s + (1.22 + 0.887i)16-s + (3.01 + 2.19i)17-s + ⋯
L(s)  = 1  + (0.608 − 0.442i)2-s + (0.993 + 0.114i)3-s + (−0.134 + 0.412i)4-s + (0.262 − 0.361i)5-s + (0.655 − 0.369i)6-s + (−0.825 − 0.268i)7-s + (0.333 + 1.02i)8-s + (0.973 + 0.227i)9-s − 0.336i·10-s + (−0.683 − 0.729i)11-s + (−0.180 + 0.394i)12-s + (−0.313 − 0.431i)13-s + (−0.620 + 0.201i)14-s + (0.302 − 0.329i)15-s + (0.305 + 0.221i)16-s + (0.732 + 0.532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.972 + 0.232i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.972 + 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83012 - 0.215548i\)
\(L(\frac12)\) \(\approx\) \(1.83012 - 0.215548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.197i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (2.26 + 2.41i)T \)
good2 \( 1 + (-0.860 + 0.625i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (2.18 + 0.709i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.13 + 1.55i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.01 - 2.19i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (8.13 - 2.64i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.81iT - 23T^{2} \)
29 \( 1 + (-1.01 + 3.13i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.98 + 2.89i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.59 - 11.0i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.619 - 1.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.20iT - 43T^{2} \)
47 \( 1 + (-3.15 + 1.02i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.55 - 4.88i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.0820 + 0.0266i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.79 + 6.60i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.92T + 67T^{2} \)
71 \( 1 + (-8.64 + 11.9i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.23 - 1.70i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.0325 - 0.0448i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.22 + 0.886i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 1.65iT - 89T^{2} \)
97 \( 1 + (-6.60 + 4.80i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.93872743160314255270487575216, −12.29645156698545987608212330199, −10.65251840357805879505672740715, −9.896257453203769170813407177181, −8.439641732151286166040821812609, −8.046704991581543433454653743963, −6.30408058921326295508296776307, −4.68520078825825901320454907858, −3.55519451897410230983832275642, −2.48863486572687035178389793690, 2.34813469759640710173394070298, 3.87114309592862183370809104655, 5.22985383632579298520119451053, 6.62431424693125182087986098445, 7.32388826736911380720761509997, 8.917696309163984007538675523513, 9.781940439246255544499635001628, 10.52206641330420614795322990086, 12.45009332048400627123217435550, 13.05019144077489566817369008621

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