↑

Properties

Label 2-165-11.3-c1-0-5
Degree $2$
Conductor $165$
Sign $0.564 + 0.825i$
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  + (1.90 − 1.38i)2-s + (0.309 + 0.951i)3-s + (1.09 − 3.37i)4-s + (−0.809 − 0.587i)5-s + (1.90 + 1.38i)6-s + (0.0598 − 0.184i)7-s + (−1.12 − 3.47i)8-s + (−0.809 + 0.587i)9-s − 2.35·10-s + (−1.96 + 2.67i)11-s + 3.54·12-s + (−0.787 + 0.572i)13-s + (−0.140 − 0.433i)14-s + (0.309 − 0.951i)15-s + (−1.21 − 0.880i)16-s + (2.16 + 1.57i)17-s + ⋯
L(s)  = 1  + (1.34 − 0.979i)2-s + (0.178 + 0.549i)3-s + (0.548 − 1.68i)4-s + (−0.361 − 0.262i)5-s + (0.778 + 0.565i)6-s + (0.0226 − 0.0695i)7-s + (−0.398 − 1.22i)8-s + (−0.269 + 0.195i)9-s − 0.744·10-s + (−0.591 + 0.806i)11-s + 1.02·12-s + (−0.218 + 0.158i)13-s + (−0.0376 − 0.115i)14-s + (0.0797 − 0.245i)15-s + (−0.303 − 0.220i)16-s + (0.525 + 0.381i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ 0.564 + 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87087 - 0.987093i\)
\(L(\frac12)\) \(\approx\) \(1.87087 - 0.987093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (1.96 - 2.67i)T \)
good2 \( 1 + (-1.90 + 1.38i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-0.0598 + 0.184i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.787 - 0.572i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.16 - 1.57i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.71 + 5.27i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.80T + 23T^{2} \)
29 \( 1 + (3.12 - 9.62i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.02 + 1.47i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.76 + 5.43i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.55 + 7.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.11T + 43T^{2} \)
47 \( 1 + (3.35 + 10.3i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.51 + 5.45i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.46 - 10.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.975 + 0.708i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.25T + 67T^{2} \)
71 \( 1 + (4.84 + 3.52i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.02 + 3.14i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.21 - 5.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.72 + 4.88i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.34T + 89T^{2} \)
97 \( 1 + (-12.8 + 9.31i)T + (29.9 - 92.2i)T^{2} \)
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

−12.62840861604668324253185950744, −11.89833356208864055005056292025, −10.82189718849627346591317122232, −10.13582852507579843179125847997, −8.822285188614603187348443268895, −7.30374377570967850289567027925, −5.58198562981746525203180816587, −4.65309934059947441333179003513, −3.69985425966542054458225769682, −2.27465340670819353207486826212, 2.90990595033121497270647484969, 4.15993027818074448142367528828, 5.63985621624884816023719803835, 6.36060329366392110630266583112, 7.72360053722438733188700399655, 8.115921650028872749056516625609, 10.01832568255525974710467851836, 11.53301337291483997580405676953, 12.30351192338643844604368302625, 13.24108365196744972660114819313

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