Properties

Label 2-165-33.8-c1-0-14
Degree $2$
Conductor $165$
Sign $0.308 + 0.951i$
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  + (1.45 − 1.05i)2-s + (0.960 − 1.44i)3-s + (0.383 − 1.17i)4-s + (−0.587 + 0.809i)5-s + (−0.125 − 3.11i)6-s + (−0.145 − 0.0472i)7-s + (0.422 + 1.30i)8-s + (−1.15 − 2.76i)9-s + 1.80i·10-s + (−3.24 + 0.670i)11-s + (−1.33 − 1.68i)12-s + (1.93 + 2.66i)13-s + (−0.261 + 0.0850i)14-s + (0.600 + 1.62i)15-s + (3.99 + 2.90i)16-s + (−4.27 − 3.10i)17-s + ⋯
L(s)  = 1  + (1.02 − 0.748i)2-s + (0.554 − 0.831i)3-s + (0.191 − 0.589i)4-s + (−0.262 + 0.361i)5-s + (−0.0511 − 1.27i)6-s + (−0.0549 − 0.0178i)7-s + (0.149 + 0.459i)8-s + (−0.384 − 0.923i)9-s + 0.569i·10-s + (−0.979 + 0.202i)11-s + (−0.384 − 0.486i)12-s + (0.536 + 0.738i)13-s + (−0.0699 + 0.0227i)14-s + (0.155 + 0.419i)15-s + (0.999 + 0.726i)16-s + (−1.03 − 0.754i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.308 + 0.951i$
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: \((2,\ 165,\ (\ :1/2),\ 0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60262 - 1.16538i\)
\(L(\frac12)\) \(\approx\) \(1.60262 - 1.16538i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.960 + 1.44i)T \)
5 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (3.24 - 0.670i)T \)
good2 \( 1 + (-1.45 + 1.05i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.145 + 0.0472i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.93 - 2.66i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.27 + 3.10i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.650 + 0.211i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.40iT - 23T^{2} \)
29 \( 1 + (-0.713 + 2.19i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.14 + 4.46i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.02 + 6.22i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.545 - 1.67i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.03iT - 43T^{2} \)
47 \( 1 + (3.53 - 1.14i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.80 + 10.7i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-8.84 - 2.87i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.87 - 3.95i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + (-1.79 + 2.47i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-8.83 - 2.87i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.524 + 0.721i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.29 + 6.02i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 4.99iT - 89T^{2} \)
97 \( 1 + (-6.77 + 4.92i)T + (29.9 - 92.2i)T^{2} \)
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

−12.92134186940479658938805352991, −11.62832280783410231319614450834, −11.28246614205615285732045088021, −9.707191622065919976915339349066, −8.354930621420735996332609683868, −7.38810050003176874384957185887, −6.11559594700996798262555891428, −4.57268805364555485416477680344, −3.24771287818072613906896535194, −2.16375918834873832300581699720, 3.08935167828216659956622958491, 4.36357434456403798577698282375, 5.18784683146681340874964646644, 6.38502054694452429280811367974, 7.914039248808011158163414135612, 8.681605103627326576061907048361, 10.13646075752582119675471566664, 10.86722673785195488176004049742, 12.51147305560301771443658616132, 13.28939888076646931315835455267

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