↑

Properties

Label 2-165-15.2-c1-0-14
Degree $2$
Conductor $165$
Sign $0.0618 + 0.998i$
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  + (1.70 − 1.70i)2-s + (−1.41 + i)3-s − 3.82i·4-s + (2 − i)5-s + (−0.707 + 4.12i)6-s + (−0.585 − 0.585i)7-s + (−3.12 − 3.12i)8-s + (1.00 − 2.82i)9-s + (1.70 − 5.12i)10-s + i·11-s + (3.82 + 5.41i)12-s + (−2 + 2i)13-s − 2·14-s + (−1.82 + 3.41i)15-s − 2.99·16-s + (−2.82 + 2.82i)17-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)2-s + (−0.816 + 0.577i)3-s − 1.91i·4-s + (0.894 − 0.447i)5-s + (−0.288 + 1.68i)6-s + (−0.221 − 0.221i)7-s + (−1.10 − 1.10i)8-s + (0.333 − 0.942i)9-s + (0.539 − 1.61i)10-s + 0.301i·11-s + (1.10 + 1.56i)12-s + (−0.554 + 0.554i)13-s − 0.534·14-s + (−0.472 + 0.881i)15-s − 0.749·16-s + (−0.685 + 0.685i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.0618 + 0.998i$
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: \((2,\ 165,\ (\ :1/2),\ 0.0618 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27408 - 1.19756i\)
\(L(\frac12)\) \(\approx\) \(1.27408 - 1.19756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.41 - i)T \)
5 \( 1 + (-2 + i)T \)
11 \( 1 - iT \)
good2 \( 1 + (-1.70 + 1.70i)T - 2iT^{2} \)
7 \( 1 + (0.585 + 0.585i)T + 7iT^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (2.82 - 2.82i)T - 17iT^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 + (-5.24 - 5.24i)T + 23iT^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 + (5.82 + 5.82i)T + 37iT^{2} \)
41 \( 1 + 3.65iT - 41T^{2} \)
43 \( 1 + (8.24 - 8.24i)T - 43iT^{2} \)
47 \( 1 + (1.24 - 1.24i)T - 47iT^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + (3.58 + 3.58i)T + 67iT^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 5.17iT - 79T^{2} \)
83 \( 1 + (5.07 + 5.07i)T + 83iT^{2} \)
89 \( 1 + 1.65T + 89T^{2} \)
97 \( 1 + (0.656 + 0.656i)T + 97iT^{2} \)
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

−12.62096298448346985626645941148, −11.70356723579755531826064168947, −10.75725677544342688778920388765, −10.00558278520824681637798998711, −9.160309333310669805326686700683, −6.71320526644074293644016264076, −5.54897928764172166104511268328, −4.78985533979576063557190331700, −3.62601729158310636448845626749, −1.76212302387421335193808868964, 2.79723636372049913452562751335, 4.87228814633053231778056462216, 5.55029364221438326140343239285, 6.70893430059244785274100604704, 7.06084094703565257204520584624, 8.614717978454643618865662403962, 10.23594814244312409302616827286, 11.38851852084243969074346068255, 12.59527411713129723278947965948, 13.17435659507405833420412985649

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