Properties

Label 2-165-1.1-c5-0-20
Degree $2$
Conductor $165$
Sign $-1$
Analytic cond. $26.4633$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.694·2-s − 9·3-s − 31.5·4-s + 25·5-s − 6.25·6-s + 83.1·7-s − 44.1·8-s + 81·9-s + 17.3·10-s + 121·11-s + 283.·12-s − 674.·13-s + 57.7·14-s − 225·15-s + 977.·16-s + 1.92e3·17-s + 56.2·18-s − 149.·19-s − 787.·20-s − 748.·21-s + 84.0·22-s − 1.35e3·23-s + 397.·24-s + 625·25-s − 468.·26-s − 729·27-s − 2.62e3·28-s + ⋯
L(s)  = 1  + 0.122·2-s − 0.577·3-s − 0.984·4-s + 0.447·5-s − 0.0709·6-s + 0.641·7-s − 0.243·8-s + 0.333·9-s + 0.0549·10-s + 0.301·11-s + 0.568·12-s − 1.10·13-s + 0.0787·14-s − 0.258·15-s + 0.954·16-s + 1.61·17-s + 0.0409·18-s − 0.0951·19-s − 0.440·20-s − 0.370·21-s + 0.0370·22-s − 0.534·23-s + 0.140·24-s + 0.200·25-s − 0.135·26-s − 0.192·27-s − 0.631·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(165\)    =    \(3 \cdot 5 \cdot 11\)
Sign: $-1$
Analytic conductor: \(26.4633\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 165,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9T \)
5 \( 1 - 25T \)
11 \( 1 - 121T \)
good2 \( 1 - 0.694T + 32T^{2} \)
7 \( 1 - 83.1T + 1.68e4T^{2} \)
13 \( 1 + 674.T + 3.71e5T^{2} \)
17 \( 1 - 1.92e3T + 1.41e6T^{2} \)
19 \( 1 + 149.T + 2.47e6T^{2} \)
23 \( 1 + 1.35e3T + 6.43e6T^{2} \)
29 \( 1 + 7.32e3T + 2.05e7T^{2} \)
31 \( 1 + 4.21e3T + 2.86e7T^{2} \)
37 \( 1 + 1.34e4T + 6.93e7T^{2} \)
41 \( 1 - 2.86e3T + 1.15e8T^{2} \)
43 \( 1 + 2.20e4T + 1.47e8T^{2} \)
47 \( 1 + 1.45e4T + 2.29e8T^{2} \)
53 \( 1 - 1.33e4T + 4.18e8T^{2} \)
59 \( 1 - 4.58e4T + 7.14e8T^{2} \)
61 \( 1 - 1.89e4T + 8.44e8T^{2} \)
67 \( 1 - 6.65e3T + 1.35e9T^{2} \)
71 \( 1 + 6.10e4T + 1.80e9T^{2} \)
73 \( 1 + 1.73e4T + 2.07e9T^{2} \)
79 \( 1 - 6.16e4T + 3.07e9T^{2} \)
83 \( 1 + 6.52e4T + 3.93e9T^{2} \)
89 \( 1 + 1.09e5T + 5.58e9T^{2} \)
97 \( 1 - 8.37e4T + 8.58e9T^{2} \)
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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

−11.60461963771608919651196250169, −10.20538566570068564590945717519, −9.607877424638736412911168724111, −8.347440477958327608953965417041, −7.22066646521201092076179309306, −5.60948020799022816143088308016, −5.03436452128696154400275564748, −3.65395604422640982017472634607, −1.59484573552482972426383062529, 0, 1.59484573552482972426383062529, 3.65395604422640982017472634607, 5.03436452128696154400275564748, 5.60948020799022816143088308016, 7.22066646521201092076179309306, 8.347440477958327608953965417041, 9.607877424638736412911168724111, 10.20538566570068564590945717519, 11.60461963771608919651196250169

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