Properties

Degree $2$
Conductor $503$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.392·2-s + 0.950·3-s − 1.84·4-s − 2.28·5-s − 0.372·6-s + 2.71·7-s + 1.50·8-s − 2.09·9-s + 0.897·10-s + 1.36·11-s − 1.75·12-s − 2.93·13-s − 1.06·14-s − 2.17·15-s + 3.10·16-s − 2.61·17-s + 0.822·18-s − 7.79·19-s + 4.22·20-s + 2.57·21-s − 0.536·22-s − 2.61·23-s + 1.43·24-s + 0.230·25-s + 1.14·26-s − 4.84·27-s − 5.00·28-s + ⋯
L(s)  = 1  − 0.277·2-s + 0.548·3-s − 0.923·4-s − 1.02·5-s − 0.152·6-s + 1.02·7-s + 0.533·8-s − 0.699·9-s + 0.283·10-s + 0.412·11-s − 0.506·12-s − 0.812·13-s − 0.284·14-s − 0.561·15-s + 0.775·16-s − 0.633·17-s + 0.193·18-s − 1.78·19-s + 0.944·20-s + 0.561·21-s − 0.114·22-s − 0.544·23-s + 0.292·24-s + 0.0460·25-s + 0.225·26-s − 0.932·27-s − 0.945·28-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{503} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 503,\ (\ :1/2),\ -1)\)

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)$
bad503 \( 1 + T \)
good2 \( 1 + 0.392T + 2T^{2} \)
3 \( 1 - 0.950T + 3T^{2} \)
5 \( 1 + 2.28T + 5T^{2} \)
7 \( 1 - 2.71T + 7T^{2} \)
11 \( 1 - 1.36T + 11T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 + 2.61T + 17T^{2} \)
19 \( 1 + 7.79T + 19T^{2} \)
23 \( 1 + 2.61T + 23T^{2} \)
29 \( 1 + 0.314T + 29T^{2} \)
31 \( 1 + 7.95T + 31T^{2} \)
37 \( 1 + 4.17T + 37T^{2} \)
41 \( 1 - 6.16T + 41T^{2} \)
43 \( 1 - 0.457T + 43T^{2} \)
47 \( 1 - 7.67T + 47T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 - 0.217T + 59T^{2} \)
61 \( 1 - 7.26T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 + 16.5T + 71T^{2} \)
73 \( 1 - 2.86T + 73T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 0.0952T + 89T^{2} \)
97 \( 1 + 9.27T + 97T^{2} \)
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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

−10.53010239666383339771505347511, −9.260756767745355465147802581238, −8.591021552571008166664217770737, −8.066189830981265060248399066376, −7.21364188783771902971732629678, −5.60490906735082322958828048930, −4.43493388878886222689102822589, −3.85280977327434200316008754894, −2.14541617555194133492562465260, 0, 2.14541617555194133492562465260, 3.85280977327434200316008754894, 4.43493388878886222689102822589, 5.60490906735082322958828048930, 7.21364188783771902971732629678, 8.066189830981265060248399066376, 8.591021552571008166664217770737, 9.260756767745355465147802581238, 10.53010239666383339771505347511

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