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Properties

Label 2-165-33.8-c1-0-15
Degree $2$
Conductor $165$
Sign $-0.147 + 0.989i$
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  + (1.54 − 1.12i)2-s + (−0.613 − 1.61i)3-s + (0.510 − 1.57i)4-s + (0.587 − 0.809i)5-s + (−2.76 − 1.81i)6-s + (−0.0795 − 0.0258i)7-s + (0.204 + 0.630i)8-s + (−2.24 + 1.98i)9-s − 1.91i·10-s + (1.17 + 3.10i)11-s + (−2.86 + 0.137i)12-s + (−3.74 − 5.14i)13-s + (−0.152 + 0.0493i)14-s + (−1.67 − 0.455i)15-s + (3.69 + 2.68i)16-s + (4.03 + 2.93i)17-s + ⋯
L(s)  = 1  + (1.09 − 0.794i)2-s + (−0.354 − 0.935i)3-s + (0.255 − 0.786i)4-s + (0.262 − 0.361i)5-s + (−1.13 − 0.741i)6-s + (−0.0300 − 0.00976i)7-s + (0.0723 + 0.222i)8-s + (−0.749 + 0.662i)9-s − 0.604i·10-s + (0.354 + 0.934i)11-s + (−0.825 + 0.0395i)12-s + (−1.03 − 1.42i)13-s + (−0.0406 + 0.0131i)14-s + (−0.431 − 0.117i)15-s + (0.924 + 0.672i)16-s + (0.978 + 0.710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-0.147 + 0.989i$
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: \((2,\ 165,\ (\ :1/2),\ -0.147 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14359 - 1.32697i\)
\(L(\frac12)\) \(\approx\) \(1.14359 - 1.32697i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.613 + 1.61i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-1.17 - 3.10i)T \)
good2 \( 1 + (-1.54 + 1.12i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.0795 + 0.0258i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.74 + 5.14i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.03 - 2.93i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.93 + 0.629i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.16iT - 23T^{2} \)
29 \( 1 + (1.70 - 5.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (6.10 - 4.43i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.18 + 3.63i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.74 + 5.36i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.31iT - 43T^{2} \)
47 \( 1 + (3.31 - 1.07i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.50 + 7.57i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.486 + 0.158i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.69 + 5.09i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.39T + 67T^{2} \)
71 \( 1 + (8.70 - 11.9i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.06 + 0.995i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.50 + 4.82i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.66 - 1.21i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 3.88iT - 89T^{2} \)
97 \( 1 + (-1.00 + 0.727i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.72883149765255317914190638342, −12.04317924333088712299852446160, −10.90520083455286868213544726176, −9.898083555145763862697370172620, −8.224620648311021406427447928620, −7.20209535113601591884278163608, −5.64884921170128558848699820218, −4.95817127491587004853542639678, −3.19148289210797760858016081356, −1.73907222578884704320130411259, 3.28955342563059083753040347127, 4.44736940815278269053639856889, 5.50624425124676902789522483237, 6.35601863326635270913348040392, 7.51483015428015906255670727257, 9.298207714726266922622394275069, 9.952575770503503860385618628703, 11.39309985830951757922810134357, 12.04356838295219166235652744674, 13.51972222999102252606613945860

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