Properties

Label 2-5290-1.1-c1-0-160
Degree $2$
Conductor $5290$
Sign $-1$
Analytic cond. $42.2408$
Root an. cond. $6.49929$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.73·3-s + 4-s − 5-s + 1.73·6-s + 0.732·7-s + 8-s − 10-s − 3.46·11-s + 1.73·12-s + 0.732·14-s − 1.73·15-s + 16-s − 3.73·17-s − 4.46·19-s − 20-s + 1.26·21-s − 3.46·22-s + 1.73·24-s + 25-s − 5.19·27-s + 0.732·28-s − 0.732·29-s − 1.73·30-s + 0.196·31-s + 32-s − 5.99·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.447·5-s + 0.707·6-s + 0.276·7-s + 0.353·8-s − 0.316·10-s − 1.04·11-s + 0.500·12-s + 0.195·14-s − 0.447·15-s + 0.250·16-s − 0.905·17-s − 1.02·19-s − 0.223·20-s + 0.276·21-s − 0.738·22-s + 0.353·24-s + 0.200·25-s − 1.00·27-s + 0.138·28-s − 0.135·29-s − 0.316·30-s + 0.0352·31-s + 0.176·32-s − 1.04·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5290\)    =    \(2 \cdot 5 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(42.2408\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5290,\ (\ :1/2),\ -1)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
23 \( 1 \)
good3 \( 1 - 1.73T + 3T^{2} \)
7 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
29 \( 1 + 0.732T + 29T^{2} \)
31 \( 1 - 0.196T + 31T^{2} \)
37 \( 1 + 4.53T + 37T^{2} \)
41 \( 1 + 8.39T + 41T^{2} \)
43 \( 1 + 9.19T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 - 8.19T + 53T^{2} \)
59 \( 1 + 5.92T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 1.73T + 67T^{2} \)
71 \( 1 - 5.26T + 71T^{2} \)
73 \( 1 + 1.19T + 73T^{2} \)
79 \( 1 - 2.19T + 79T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 - 6.39T + 97T^{2} \)
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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

−7.889380013817030033532315537819, −7.17654556680826049700471754824, −6.45787930808684760345840156059, −5.48998804059451586097240085382, −4.81354510472254401576218933202, −4.02089084361273029699273033823, −3.29494968027055432555737442050, −2.50811855396218150657590962044, −1.85762107957747586034295888855, 0, 1.85762107957747586034295888855, 2.50811855396218150657590962044, 3.29494968027055432555737442050, 4.02089084361273029699273033823, 4.81354510472254401576218933202, 5.48998804059451586097240085382, 6.45787930808684760345840156059, 7.17654556680826049700471754824, 7.889380013817030033532315537819

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