↑

Properties

Label 2-165-15.2-c1-0-10
Degree $2$
Conductor $165$
Sign $0.681 + 0.732i$
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  + (−1.05 + 1.05i)2-s + (−1.71 + 0.215i)3-s − 0.232i·4-s + (−0.158 − 2.23i)5-s + (1.58 − 2.04i)6-s + (−1.04 − 1.04i)7-s + (−1.86 − 1.86i)8-s + (2.90 − 0.740i)9-s + (2.52 + 2.18i)10-s + i·11-s + (0.0500 + 0.399i)12-s + (3.37 − 3.37i)13-s + 2.20·14-s + (0.753 + 3.79i)15-s + 4.41·16-s + (−0.138 + 0.138i)17-s + ⋯
L(s)  = 1  + (−0.747 + 0.747i)2-s + (−0.992 + 0.124i)3-s − 0.116i·4-s + (−0.0710 − 0.997i)5-s + (0.648 − 0.834i)6-s + (−0.394 − 0.394i)7-s + (−0.660 − 0.660i)8-s + (0.969 − 0.246i)9-s + (0.798 + 0.692i)10-s + 0.301i·11-s + (0.0144 + 0.115i)12-s + (0.937 − 0.937i)13-s + 0.589·14-s + (0.194 + 0.980i)15-s + 1.10·16-s + (−0.0335 + 0.0335i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.681 + 0.732i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.681 + 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.385434 - 0.167832i\)
\(L(\frac12)\) \(\approx\) \(0.385434 - 0.167832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.215i)T \)
5 \( 1 + (0.158 + 2.23i)T \)
11 \( 1 - iT \)
good2 \( 1 + (1.05 - 1.05i)T - 2iT^{2} \)
7 \( 1 + (1.04 + 1.04i)T + 7iT^{2} \)
13 \( 1 + (-3.37 + 3.37i)T - 13iT^{2} \)
17 \( 1 + (0.138 - 0.138i)T - 17iT^{2} \)
19 \( 1 + 6.97iT - 19T^{2} \)
23 \( 1 + (2.64 + 2.64i)T + 23iT^{2} \)
29 \( 1 + 0.161T + 29T^{2} \)
31 \( 1 + 7.53T + 31T^{2} \)
37 \( 1 + (-3.94 - 3.94i)T + 37iT^{2} \)
41 \( 1 + 8.21iT - 41T^{2} \)
43 \( 1 + (1.14 - 1.14i)T - 43iT^{2} \)
47 \( 1 + (0.797 - 0.797i)T - 47iT^{2} \)
53 \( 1 + (0.0372 + 0.0372i)T + 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 8.33T + 61T^{2} \)
67 \( 1 + (-5.41 - 5.41i)T + 67iT^{2} \)
71 \( 1 - 15.1iT - 71T^{2} \)
73 \( 1 + (-3.98 + 3.98i)T - 73iT^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 + (-5.92 - 5.92i)T + 83iT^{2} \)
89 \( 1 - 2.98T + 89T^{2} \)
97 \( 1 + (-7.94 - 7.94i)T + 97iT^{2} \)
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

−12.80695965741356614195237795868, −11.71775336150983772712431714589, −10.53907499876784572204094205961, −9.501082010061520598900663538204, −8.584783468864539029008571567685, −7.44061066472444589031316904173, −6.43484842333530134008364787889, −5.31355155389589603199776516509, −3.90460482211590752401667820238, −0.58095569844627361765376916047, 1.81661796984633300115003843505, 3.66307168608580076295221975042, 5.75983851749084927792702848839, 6.38264760952786675606983248718, 7.84477447838563063235262061901, 9.304883539246839547837244986879, 10.14664167891541026787146537216, 11.04044904893813152612381537828, 11.55300798387644470439221778816, 12.50486623759697652548848393822

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