Properties

Label 2-165-1.1-c3-0-17
Degree $2$
Conductor $165$
Sign $-1$
Analytic cond. $9.73531$
Root an. cond. $3.12014$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·2-s + 3·3-s − 5.56·4-s − 5·5-s + 4.68·6-s − 10.2·7-s − 21.1·8-s + 9·9-s − 7.80·10-s − 11·11-s − 16.6·12-s − 40.8·13-s − 16·14-s − 15·15-s + 11.4·16-s − 98.7·17-s + 14.0·18-s − 39.6·19-s + 27.8·20-s − 30.7·21-s − 17.1·22-s + 61.6·23-s − 63.5·24-s + 25·25-s − 63.8·26-s + 27·27-s + 56.9·28-s + ⋯
L(s)  = 1  + 0.552·2-s + 0.577·3-s − 0.695·4-s − 0.447·5-s + 0.318·6-s − 0.553·7-s − 0.935·8-s + 0.333·9-s − 0.246·10-s − 0.301·11-s − 0.401·12-s − 0.872·13-s − 0.305·14-s − 0.258·15-s + 0.178·16-s − 1.40·17-s + 0.184·18-s − 0.478·19-s + 0.310·20-s − 0.319·21-s − 0.166·22-s + 0.559·23-s − 0.540·24-s + 0.200·25-s − 0.481·26-s + 0.192·27-s + 0.384·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(9.73531\)
Root analytic conductor: \(3.12014\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{165} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 165,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 + 5T \)
11 \( 1 + 11T \)
good2 \( 1 - 1.56T + 8T^{2} \)
7 \( 1 + 10.2T + 343T^{2} \)
13 \( 1 + 40.8T + 2.19e3T^{2} \)
17 \( 1 + 98.7T + 4.91e3T^{2} \)
19 \( 1 + 39.6T + 6.85e3T^{2} \)
23 \( 1 - 61.6T + 1.21e4T^{2} \)
29 \( 1 + 149.T + 2.43e4T^{2} \)
31 \( 1 - 54.7T + 2.97e4T^{2} \)
37 \( 1 - 44.8T + 5.06e4T^{2} \)
41 \( 1 - 336.T + 6.89e4T^{2} \)
43 \( 1 + 2.36T + 7.95e4T^{2} \)
47 \( 1 + 333.T + 1.03e5T^{2} \)
53 \( 1 - 640.T + 1.48e5T^{2} \)
59 \( 1 + 370.T + 2.05e5T^{2} \)
61 \( 1 + 714.T + 2.26e5T^{2} \)
67 \( 1 + 404.T + 3.00e5T^{2} \)
71 \( 1 - 939.T + 3.57e5T^{2} \)
73 \( 1 + 362.T + 3.89e5T^{2} \)
79 \( 1 - 951.T + 4.93e5T^{2} \)
83 \( 1 - 735.T + 5.71e5T^{2} \)
89 \( 1 - 385.T + 7.04e5T^{2} \)
97 \( 1 + 966.T + 9.12e5T^{2} \)
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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

−12.23792679064640545073851997807, −10.90404551148103832380410403804, −9.597167129720741986027704417040, −8.894333398709967078008516095039, −7.73987841290595667377532682041, −6.46857242033609109549585217379, −4.94825489858654283715416737940, −3.96145918542717960996069715914, −2.67272850484331780433849265635, 0, 2.67272850484331780433849265635, 3.96145918542717960996069715914, 4.94825489858654283715416737940, 6.46857242033609109549585217379, 7.73987841290595667377532682041, 8.894333398709967078008516095039, 9.597167129720741986027704417040, 10.90404551148103832380410403804, 12.23792679064640545073851997807

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