Properties

Label 2-1502-1.1-c1-0-48
Degree $2$
Conductor $1502$
Sign $1$
Analytic cond. $11.9935$
Root an. cond. $3.46316$
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 + 2.04·3-s + 4-s + 3.81·5-s + 2.04·6-s + 0.856·7-s + 8-s + 1.18·9-s + 3.81·10-s − 0.287·11-s + 2.04·12-s − 2.99·13-s + 0.856·14-s + 7.80·15-s + 16-s + 0.234·17-s + 1.18·18-s − 6.00·19-s + 3.81·20-s + 1.75·21-s − 0.287·22-s + 0.503·23-s + 2.04·24-s + 9.54·25-s − 2.99·26-s − 3.71·27-s + 0.856·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.18·3-s + 0.5·4-s + 1.70·5-s + 0.835·6-s + 0.323·7-s + 0.353·8-s + 0.394·9-s + 1.20·10-s − 0.0868·11-s + 0.590·12-s − 0.829·13-s + 0.229·14-s + 2.01·15-s + 0.250·16-s + 0.0567·17-s + 0.278·18-s − 1.37·19-s + 0.852·20-s + 0.382·21-s − 0.0613·22-s + 0.105·23-s + 0.417·24-s + 1.90·25-s − 0.586·26-s − 0.715·27-s + 0.161·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1502\)    =    \(2 \cdot 751\)
Sign: $1$
Analytic conductor: \(11.9935\)
Root analytic conductor: \(3.46316\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1502,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.930341277\)
\(L(\frac12)\) \(\approx\) \(4.930341277\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
751 \( 1 + T \)
good3 \( 1 - 2.04T + 3T^{2} \)
5 \( 1 - 3.81T + 5T^{2} \)
7 \( 1 - 0.856T + 7T^{2} \)
11 \( 1 + 0.287T + 11T^{2} \)
13 \( 1 + 2.99T + 13T^{2} \)
17 \( 1 - 0.234T + 17T^{2} \)
19 \( 1 + 6.00T + 19T^{2} \)
23 \( 1 - 0.503T + 23T^{2} \)
29 \( 1 + 5.92T + 29T^{2} \)
31 \( 1 - 8.72T + 31T^{2} \)
37 \( 1 + 8.74T + 37T^{2} \)
41 \( 1 - 4.43T + 41T^{2} \)
43 \( 1 - 0.190T + 43T^{2} \)
47 \( 1 + 7.85T + 47T^{2} \)
53 \( 1 - 8.87T + 53T^{2} \)
59 \( 1 + 5.29T + 59T^{2} \)
61 \( 1 + 2.70T + 61T^{2} \)
67 \( 1 - 7.01T + 67T^{2} \)
71 \( 1 - 4.28T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + 2.45T + 79T^{2} \)
83 \( 1 - 9.82T + 83T^{2} \)
89 \( 1 - 1.69T + 89T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−9.479281035650972174386306019334, −8.737388491165584220216764691105, −7.962560693145141870573553827022, −6.92813942154898554946612400158, −6.14089575991210090198098325859, −5.30942515141100329346991308412, −4.47002258462638556987917228057, −3.22044099281045178775448601626, −2.33598000808699426505603607165, −1.82029044978213020993903340065, 1.82029044978213020993903340065, 2.33598000808699426505603607165, 3.22044099281045178775448601626, 4.47002258462638556987917228057, 5.30942515141100329346991308412, 6.14089575991210090198098325859, 6.92813942154898554946612400158, 7.962560693145141870573553827022, 8.737388491165584220216764691105, 9.479281035650972174386306019334

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