Properties

Label 2-2366-1.1-c1-0-15
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 + 0.267·5-s − 1.73·6-s − 7-s + 8-s + 0.267·10-s − 5.46·11-s − 1.73·12-s − 14-s − 0.464·15-s + 16-s − 3.46·17-s + 3.46·19-s + 0.267·20-s + 1.73·21-s − 5.46·22-s + 8.46·23-s − 1.73·24-s − 4.92·25-s + 5.19·27-s − 28-s + 8.92·29-s − 0.464·30-s + 0.535·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.00·3-s + 0.5·4-s + 0.119·5-s − 0.707·6-s − 0.377·7-s + 0.353·8-s + 0.0847·10-s − 1.64·11-s − 0.500·12-s − 0.267·14-s − 0.119·15-s + 0.250·16-s − 0.840·17-s + 0.794·19-s + 0.0599·20-s + 0.377·21-s − 1.16·22-s + 1.76·23-s − 0.353·24-s − 0.985·25-s + 1.00·27-s − 0.188·28-s + 1.65·29-s − 0.0847·30-s + 0.0962·31-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

\(L(1)\) \(\approx\) \(1.525570819\)
\(L(\frac12)\) \(\approx\) \(1.525570819\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.267T + 5T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 8.46T + 23T^{2} \)
29 \( 1 - 8.92T + 29T^{2} \)
31 \( 1 - 0.535T + 31T^{2} \)
37 \( 1 + 2.53T + 37T^{2} \)
41 \( 1 - 9.46T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 4.92T + 47T^{2} \)
53 \( 1 + 6.92T + 53T^{2} \)
59 \( 1 - 2.80T + 59T^{2} \)
61 \( 1 - 3.19T + 61T^{2} \)
67 \( 1 - 4.92T + 67T^{2} \)
71 \( 1 + 2.46T + 71T^{2} \)
73 \( 1 + 0.535T + 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 + 8.53T + 89T^{2} \)
97 \( 1 + 3.07T + 97T^{2} \)
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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

−9.008217266035471692701277938706, −8.031010451297953842655382973648, −7.20353382267183665932357909923, −6.46380304407251605948649726006, −5.68512614144125951373540428970, −5.13995445666445950073389919664, −4.45832591127340565026352871409, −3.10063120846133244705537600991, −2.46727670677770489100878024671, −0.72705306368052666882458836178, 0.72705306368052666882458836178, 2.46727670677770489100878024671, 3.10063120846133244705537600991, 4.45832591127340565026352871409, 5.13995445666445950073389919664, 5.68512614144125951373540428970, 6.46380304407251605948649726006, 7.20353382267183665932357909923, 8.031010451297953842655382973648, 9.008217266035471692701277938706

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