Properties

Label 2-2366-1.1-c1-0-16
Degree $2$
Conductor $2366$
Sign $1$
Analytic cond. $18.8926$
Root an. cond. $4.34656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.73·3-s + 4-s − 3.73·5-s − 1.73·6-s + 7-s − 8-s + 3.73·10-s − 1.46·11-s + 1.73·12-s − 14-s − 6.46·15-s + 16-s + 3.46·17-s + 3.46·19-s − 3.73·20-s + 1.73·21-s + 1.46·22-s + 1.53·23-s − 1.73·24-s + 8.92·25-s − 5.19·27-s + 28-s − 4.92·29-s + 6.46·30-s − 7.46·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.00·3-s + 0.5·4-s − 1.66·5-s − 0.707·6-s + 0.377·7-s − 0.353·8-s + 1.18·10-s − 0.441·11-s + 0.500·12-s − 0.267·14-s − 1.66·15-s + 0.250·16-s + 0.840·17-s + 0.794·19-s − 0.834·20-s + 0.377·21-s + 0.312·22-s + 0.320·23-s − 0.353·24-s + 1.78·25-s − 1.00·27-s + 0.188·28-s − 0.915·29-s + 1.18·30-s − 1.34·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2366\)    =    \(2 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(18.8926\)
Root analytic conductor: \(4.34656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2366,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.190780746\)
\(L(\frac12)\) \(\approx\) \(1.190780746\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 \)
good3 \( 1 - 1.73T + 3T^{2} \)
5 \( 1 + 3.73T + 5T^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 1.53T + 23T^{2} \)
29 \( 1 + 4.92T + 29T^{2} \)
31 \( 1 + 7.46T + 31T^{2} \)
37 \( 1 - 9.46T + 37T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 8.92T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 + 7.19T + 61T^{2} \)
67 \( 1 - 8.92T + 67T^{2} \)
71 \( 1 + 4.46T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−8.985482468481825860122979382925, −8.014670133378411148586388291917, −7.65111527987991316272985377897, −7.35849111478084463727392316474, −5.92066937541887510734909664510, −4.90396048904913994711295690737, −3.73706899370529008336117419723, −3.27772453528571475243015271567, −2.21925446904970875547353981392, −0.73042288384820641168491640541, 0.73042288384820641168491640541, 2.21925446904970875547353981392, 3.27772453528571475243015271567, 3.73706899370529008336117419723, 4.90396048904913994711295690737, 5.92066937541887510734909664510, 7.35849111478084463727392316474, 7.65111527987991316272985377897, 8.014670133378411148586388291917, 8.985482468481825860122979382925

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