Properties

Label 2-165-1.1-c3-0-13
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  − 2.56·2-s + 3·3-s − 1.43·4-s − 5·5-s − 7.68·6-s + 6.24·7-s + 24.1·8-s + 9·9-s + 12.8·10-s − 11·11-s − 4.31·12-s − 49.1·13-s − 16·14-s − 15·15-s − 50.4·16-s + 82.7·17-s − 23.0·18-s − 130.·19-s + 7.19·20-s + 18.7·21-s + 28.1·22-s − 185.·23-s + 72.5·24-s + 25·25-s + 125.·26-s + 27·27-s − 8.98·28-s + ⋯
L(s)  = 1  − 0.905·2-s + 0.577·3-s − 0.179·4-s − 0.447·5-s − 0.522·6-s + 0.337·7-s + 1.06·8-s + 0.333·9-s + 0.405·10-s − 0.301·11-s − 0.103·12-s − 1.04·13-s − 0.305·14-s − 0.258·15-s − 0.787·16-s + 1.17·17-s − 0.301·18-s − 1.57·19-s + 0.0804·20-s + 0.194·21-s + 0.273·22-s − 1.68·23-s + 0.616·24-s + 0.200·25-s + 0.949·26-s + 0.192·27-s − 0.0606·28-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 \( 1 + 2.56T + 8T^{2} \)
7 \( 1 - 6.24T + 343T^{2} \)
13 \( 1 + 49.1T + 2.19e3T^{2} \)
17 \( 1 - 82.7T + 4.91e3T^{2} \)
19 \( 1 + 130.T + 6.85e3T^{2} \)
23 \( 1 + 185.T + 1.21e4T^{2} \)
29 \( 1 + 8.90T + 2.43e4T^{2} \)
31 \( 1 - 5.26T + 2.97e4T^{2} \)
37 \( 1 + 416.T + 5.06e4T^{2} \)
41 \( 1 + 298.T + 6.89e4T^{2} \)
43 \( 1 + 513.T + 7.95e4T^{2} \)
47 \( 1 - 557.T + 1.03e5T^{2} \)
53 \( 1 + 168.T + 1.48e5T^{2} \)
59 \( 1 - 618.T + 2.05e5T^{2} \)
61 \( 1 - 786.T + 2.26e5T^{2} \)
67 \( 1 + 339.T + 3.00e5T^{2} \)
71 \( 1 - 1.12e3T + 3.57e5T^{2} \)
73 \( 1 + 123.T + 3.89e5T^{2} \)
79 \( 1 + 309.T + 4.93e5T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 + 141.T + 7.04e5T^{2} \)
97 \( 1 - 798.T + 9.12e5T^{2} \)
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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

−11.85094862469132100314779326842, −10.37622224819429722673356132702, −9.908130172095537346889267006193, −8.541040759243852194573333948699, −8.072731700173424179396919041230, −7.03109962557393122411992906109, −5.09468681984002093006600297473, −3.84880495187668241124743405475, −1.95852522855644175105115195687, 0, 1.95852522855644175105115195687, 3.84880495187668241124743405475, 5.09468681984002093006600297473, 7.03109962557393122411992906109, 8.072731700173424179396919041230, 8.541040759243852194573333948699, 9.908130172095537346889267006193, 10.37622224819429722673356132702, 11.85094862469132100314779326842

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