Properties

Label 2-165-1.1-c3-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.63·2-s + 3·3-s + 5.23·4-s + 5·5-s + 10.9·6-s + 20.8·7-s − 10.0·8-s + 9·9-s + 18.1·10-s − 11·11-s + 15.7·12-s + 67.4·13-s + 75.8·14-s + 15·15-s − 78.4·16-s − 57.8·17-s + 32.7·18-s − 7.98·19-s + 26.1·20-s + 62.5·21-s − 40.0·22-s + 67.5·23-s − 30.1·24-s + 25·25-s + 245.·26-s + 27·27-s + 109.·28-s + ⋯
L(s)  = 1  + 1.28·2-s + 0.577·3-s + 0.654·4-s + 0.447·5-s + 0.742·6-s + 1.12·7-s − 0.444·8-s + 0.333·9-s + 0.575·10-s − 0.301·11-s + 0.377·12-s + 1.44·13-s + 1.44·14-s + 0.258·15-s − 1.22·16-s − 0.824·17-s + 0.428·18-s − 0.0964·19-s + 0.292·20-s + 0.649·21-s − 0.387·22-s + 0.612·23-s − 0.256·24-s + 0.200·25-s + 1.85·26-s + 0.192·27-s + 0.736·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 165,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.214410287\)
\(L(\frac12)\) \(\approx\) \(4.214410287\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.63T + 8T^{2} \)
7 \( 1 - 20.8T + 343T^{2} \)
13 \( 1 - 67.4T + 2.19e3T^{2} \)
17 \( 1 + 57.8T + 4.91e3T^{2} \)
19 \( 1 + 7.98T + 6.85e3T^{2} \)
23 \( 1 - 67.5T + 1.21e4T^{2} \)
29 \( 1 + 56.1T + 2.43e4T^{2} \)
31 \( 1 + 127.T + 2.97e4T^{2} \)
37 \( 1 + 95.4T + 5.06e4T^{2} \)
41 \( 1 + 485.T + 6.89e4T^{2} \)
43 \( 1 + 146.T + 7.95e4T^{2} \)
47 \( 1 - 164.T + 1.03e5T^{2} \)
53 \( 1 - 431.T + 1.48e5T^{2} \)
59 \( 1 + 804.T + 2.05e5T^{2} \)
61 \( 1 + 120.T + 2.26e5T^{2} \)
67 \( 1 + 371.T + 3.00e5T^{2} \)
71 \( 1 - 529.T + 3.57e5T^{2} \)
73 \( 1 - 1.05e3T + 3.89e5T^{2} \)
79 \( 1 + 168.T + 4.93e5T^{2} \)
83 \( 1 + 144.T + 5.71e5T^{2} \)
89 \( 1 + 1.40e3T + 7.04e5T^{2} \)
97 \( 1 - 29.6T + 9.12e5T^{2} \)
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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

−12.74067762840630571975510212798, −11.50141604019984832815038079111, −10.69908882793999199367807821388, −9.089922618354779870118949989772, −8.329606273408107556272876338666, −6.81258652239268613973793372121, −5.58365213359824708809163172614, −4.57919032015402283640228629244, −3.38921215744865041769093438211, −1.86464315129914068725919318431, 1.86464315129914068725919318431, 3.38921215744865041769093438211, 4.57919032015402283640228629244, 5.58365213359824708809163172614, 6.81258652239268613973793372121, 8.329606273408107556272876338666, 9.089922618354779870118949989772, 10.69908882793999199367807821388, 11.50141604019984832815038079111, 12.74067762840630571975510212798

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