Properties

Label 2-503-1.1-c1-0-20
Degree $2$
Conductor $503$
Sign $1$
Analytic cond. $4.01647$
Root an. cond. $2.00411$
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  − 1.43·2-s + 2.32·3-s + 0.0487·4-s + 1.62·5-s − 3.33·6-s + 3.43·7-s + 2.79·8-s + 2.42·9-s − 2.32·10-s − 0.695·11-s + 0.113·12-s + 2.78·13-s − 4.91·14-s + 3.77·15-s − 4.09·16-s − 6.87·17-s − 3.46·18-s − 1.23·19-s + 0.0790·20-s + 7.99·21-s + 0.995·22-s + 8.03·23-s + 6.50·24-s − 2.37·25-s − 3.98·26-s − 1.34·27-s + 0.167·28-s + ⋯
L(s)  = 1  − 1.01·2-s + 1.34·3-s + 0.0243·4-s + 0.725·5-s − 1.36·6-s + 1.29·7-s + 0.987·8-s + 0.806·9-s − 0.733·10-s − 0.209·11-s + 0.0327·12-s + 0.772·13-s − 1.31·14-s + 0.974·15-s − 1.02·16-s − 1.66·17-s − 0.816·18-s − 0.282·19-s + 0.0176·20-s + 1.74·21-s + 0.212·22-s + 1.67·23-s + 1.32·24-s − 0.474·25-s − 0.782·26-s − 0.259·27-s + 0.0316·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(503\)
Sign: $1$
Analytic conductor: \(4.01647\)
Root analytic conductor: \(2.00411\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 503,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.539510113\)
\(L(\frac12)\) \(\approx\) \(1.539510113\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad503 \( 1 - T \)
good2 \( 1 + 1.43T + 2T^{2} \)
3 \( 1 - 2.32T + 3T^{2} \)
5 \( 1 - 1.62T + 5T^{2} \)
7 \( 1 - 3.43T + 7T^{2} \)
11 \( 1 + 0.695T + 11T^{2} \)
13 \( 1 - 2.78T + 13T^{2} \)
17 \( 1 + 6.87T + 17T^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
23 \( 1 - 8.03T + 23T^{2} \)
29 \( 1 + 8.02T + 29T^{2} \)
31 \( 1 - 9.59T + 31T^{2} \)
37 \( 1 + 0.0454T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 - 1.72T + 47T^{2} \)
53 \( 1 + 6.66T + 53T^{2} \)
59 \( 1 + 7.83T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 - 0.128T + 67T^{2} \)
71 \( 1 - 9.36T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 9.03T + 79T^{2} \)
83 \( 1 + 0.939T + 83T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−10.85476611607895665169626607505, −9.591735838233713224268900275092, −9.085031003426995824724393530555, −8.347662195588282184829153474839, −7.87136460936461021509207125846, −6.68807324606357218789201863886, −5.10986974185637095655779923254, −4.09630177157712334487693746332, −2.43794511155899922805173577606, −1.52601554781974830404929905906, 1.52601554781974830404929905906, 2.43794511155899922805173577606, 4.09630177157712334487693746332, 5.10986974185637095655779923254, 6.68807324606357218789201863886, 7.87136460936461021509207125846, 8.347662195588282184829153474839, 9.085031003426995824724393530555, 9.591735838233713224268900275092, 10.85476611607895665169626607505

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