Properties

Label 2-1006-1.1-c1-0-36
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s − 2·9-s − 5·11-s − 12-s − 5·13-s + 14-s + 16-s − 2·18-s − 6·19-s − 21-s − 5·22-s + 23-s − 24-s − 5·25-s − 5·26-s + 5·27-s + 28-s + 6·29-s + 8·31-s + 32-s + 5·33-s − 2·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 1.50·11-s − 0.288·12-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.471·18-s − 1.37·19-s − 0.218·21-s − 1.06·22-s + 0.208·23-s − 0.204·24-s − 25-s − 0.980·26-s + 0.962·27-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.870·33-s − 1/3·36-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: \(1\)
Selberg data: \((2,\ 1006,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
503 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.919580205497809214721828176561, −8.426981763238328825777480614947, −7.902476972027407870307469853024, −6.79642318501152767639984710260, −5.98297584186990174254768513700, −5.00558535884058916147640308815, −4.65621481774096026687557687085, −3.02811664827220844560879586853, −2.20099778339682437421938450648, 0, 2.20099778339682437421938450648, 3.02811664827220844560879586853, 4.65621481774096026687557687085, 5.00558535884058916147640308815, 5.98297584186990174254768513700, 6.79642318501152767639984710260, 7.902476972027407870307469853024, 8.426981763238328825777480614947, 9.919580205497809214721828176561

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