Properties

Label 2-1502-1.1-c1-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3.37·3-s + 4-s − 4.42·5-s − 3.37·6-s + 1.01·7-s + 8-s + 8.35·9-s − 4.42·10-s + 3.10·11-s − 3.37·12-s − 4.40·13-s + 1.01·14-s + 14.9·15-s + 16-s + 0.837·17-s + 8.35·18-s + 6.78·19-s − 4.42·20-s − 3.41·21-s + 3.10·22-s + 2.91·23-s − 3.37·24-s + 14.5·25-s − 4.40·26-s − 18.0·27-s + 1.01·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.94·3-s + 0.5·4-s − 1.97·5-s − 1.37·6-s + 0.382·7-s + 0.353·8-s + 2.78·9-s − 1.39·10-s + 0.937·11-s − 0.972·12-s − 1.22·13-s + 0.270·14-s + 3.85·15-s + 0.250·16-s + 0.203·17-s + 1.97·18-s + 1.55·19-s − 0.989·20-s − 0.744·21-s + 0.662·22-s + 0.607·23-s − 0.687·24-s + 2.91·25-s − 0.863·26-s − 3.47·27-s + 0.191·28-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: \(1\)
Selberg data: \((2,\ 1502,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
751 \( 1 - T \)
good3 \( 1 + 3.37T + 3T^{2} \)
5 \( 1 + 4.42T + 5T^{2} \)
7 \( 1 - 1.01T + 7T^{2} \)
11 \( 1 - 3.10T + 11T^{2} \)
13 \( 1 + 4.40T + 13T^{2} \)
17 \( 1 - 0.837T + 17T^{2} \)
19 \( 1 - 6.78T + 19T^{2} \)
23 \( 1 - 2.91T + 23T^{2} \)
29 \( 1 + 6.10T + 29T^{2} \)
31 \( 1 + 6.14T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 - 6.65T + 41T^{2} \)
43 \( 1 + 7.74T + 43T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 - 0.695T + 59T^{2} \)
61 \( 1 - 4.68T + 61T^{2} \)
67 \( 1 + 4.54T + 67T^{2} \)
71 \( 1 + 7.73T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 + 2.20T + 79T^{2} \)
83 \( 1 - 6.21T + 83T^{2} \)
89 \( 1 + 0.410T + 89T^{2} \)
97 \( 1 + 0.409T + 97T^{2} \)
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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

−9.200934256292264774349275238660, −7.73458894964343075584090940464, −7.19645341052473012959156940382, −6.79891352509249359549540018453, −5.38769415965709949050655266253, −5.04111776769349297975708012291, −4.15214027400400402246162605159, −3.47780742911357661011785047431, −1.30933912099619989462661710760, 0, 1.30933912099619989462661710760, 3.47780742911357661011785047431, 4.15214027400400402246162605159, 5.04111776769349297975708012291, 5.38769415965709949050655266253, 6.79891352509249359549540018453, 7.19645341052473012959156940382, 7.73458894964343075584090940464, 9.200934256292264774349275238660

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