Properties

Label 2-1006-1.1-c1-0-6
Degree $2$
Conductor $1006$
Sign $1$
Analytic cond. $8.03295$
Root an. cond. $2.83424$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 3·9-s + 4·11-s + 2·13-s + 16-s − 2·17-s + 3·18-s − 2·19-s − 4·22-s + 8·23-s − 5·25-s − 2·26-s + 8·31-s − 32-s + 2·34-s − 3·36-s + 8·37-s + 2·38-s + 6·41-s + 4·43-s + 4·44-s − 8·46-s − 8·47-s − 7·49-s + 5·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.458·19-s − 0.852·22-s + 1.66·23-s − 25-s − 0.392·26-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.31·37-s + 0.324·38-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 1.16·47-s − 49-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1006\)    =    \(2 \cdot 503\)
Sign: $1$
Analytic conductor: \(8.03295\)
Root analytic conductor: \(2.83424\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1006,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.138501340\)
\(L(\frac12)\) \(\approx\) \(1.138501340\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
503 \( 1 - T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−9.776853377242166714321641509123, −9.045713520834134029024985085465, −8.518177580053212777301486775367, −7.60149755922095367584842945485, −6.48129940793107102458025933478, −6.05310818168592297446874013690, −4.69049447172712355583991760980, −3.51782401576178792422064914327, −2.39892043457039317717883491800, −0.946257334458773807700757924424, 0.946257334458773807700757924424, 2.39892043457039317717883491800, 3.51782401576178792422064914327, 4.69049447172712355583991760980, 6.05310818168592297446874013690, 6.48129940793107102458025933478, 7.60149755922095367584842945485, 8.518177580053212777301486775367, 9.045713520834134029024985085465, 9.776853377242166714321641509123

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