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Properties

Label 2-165-33.29-c1-0-14
Degree $2$
Conductor $165$
Sign $-0.368 + 0.929i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.453 − 0.329i)2-s + (1.08 − 1.34i)3-s + (−0.520 − 1.60i)4-s + (−0.587 − 0.809i)5-s + (−0.938 + 0.253i)6-s + (−0.254 + 0.0827i)7-s + (−0.638 + 1.96i)8-s + (−0.633 − 2.93i)9-s + 0.561i·10-s + (−1.79 + 2.79i)11-s + (−2.72 − 1.04i)12-s + (1.44 − 1.99i)13-s + (0.142 + 0.0464i)14-s + (−1.72 − 0.0877i)15-s + (−1.78 + 1.29i)16-s + (3.05 − 2.21i)17-s + ⋯
L(s)  = 1  + (−0.320 − 0.233i)2-s + (0.628 − 0.778i)3-s + (−0.260 − 0.801i)4-s + (−0.262 − 0.361i)5-s + (−0.383 + 0.103i)6-s + (−0.0962 + 0.0312i)7-s + (−0.225 + 0.695i)8-s + (−0.211 − 0.977i)9-s + 0.177i·10-s + (−0.540 + 0.841i)11-s + (−0.787 − 0.300i)12-s + (0.401 − 0.552i)13-s + (0.0381 + 0.0124i)14-s + (−0.446 − 0.0226i)15-s + (−0.447 + 0.324i)16-s + (0.739 − 0.537i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.368 + 0.929i$
Analytic conductor: \(1.31753\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :1/2),\ -0.368 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.575165 - 0.846439i\)
\(L(\frac12)\) \(\approx\) \(0.575165 - 0.846439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.08 + 1.34i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (1.79 - 2.79i)T \)
good2 \( 1 + (0.453 + 0.329i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.254 - 0.0827i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.44 + 1.99i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.05 + 2.21i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-6.82 - 2.21i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.58iT - 23T^{2} \)
29 \( 1 + (0.678 + 2.08i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.54 - 4.03i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.374 - 1.15i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.83 - 11.7i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.181iT - 43T^{2} \)
47 \( 1 + (-9.04 - 2.93i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.15 + 1.58i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (5.22 - 1.69i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.499 + 0.687i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.419T + 67T^{2} \)
71 \( 1 + (1.47 + 2.03i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (7.26 - 2.35i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.27 + 4.50i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.09 + 1.52i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 7.27iT - 89T^{2} \)
97 \( 1 + (15.1 + 10.9i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.51547750001501539544775673587, −11.69122798779752364425686036302, −10.25156445329901204853542702692, −9.517461115876008451936009177693, −8.390790889341069923699113590437, −7.54382782638550141179210285895, −6.12427895577139114232053061655, −4.84632929809680210135768675375, −2.89145199503039739935759108256, −1.14972015560171559296756924655, 3.07658394696877312086685365540, 3.86625491693487852988492947654, 5.48243744541671767660814609999, 7.26303337969206975593885927353, 8.069640920086881344988766371751, 9.007597070666434994832928249000, 9.889011989346464082732478903912, 11.07286237193189498215557481949, 12.02644890303101025902525553467, 13.54676685119979292121202211619

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