Properties

Label 2-503-503.2-c1-0-5
Degree $2$
Conductor $503$
Sign $0.999 + 0.0212i$
Analytic cond. $4.01647$
Root an. cond. $2.00411$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.72 − 2.12i)2-s + (1.69 − 1.47i)3-s + (−1.11 + 5.34i)4-s + (−0.0846 − 0.267i)5-s + (−6.04 − 1.06i)6-s + (−1.43 + 4.15i)7-s + (8.40 − 4.32i)8-s + (0.294 − 2.02i)9-s + (−0.421 + 0.640i)10-s + (−3.38 + 1.09i)11-s + (5.95 + 10.7i)12-s + (−4.38 + 1.47i)13-s + (11.2 − 4.12i)14-s + (−0.537 − 0.329i)15-s + (−13.6 − 5.97i)16-s + (0.274 + 0.137i)17-s + ⋯
L(s)  = 1  + (−1.21 − 1.49i)2-s + (0.981 − 0.849i)3-s + (−0.559 + 2.67i)4-s + (−0.0378 − 0.119i)5-s + (−2.46 − 0.436i)6-s + (−0.540 + 1.57i)7-s + (2.96 − 1.52i)8-s + (0.0980 − 0.676i)9-s + (−0.133 + 0.202i)10-s + (−1.01 + 0.329i)11-s + (1.71 + 3.09i)12-s + (−1.21 + 0.410i)13-s + (3.01 − 1.10i)14-s + (−0.138 − 0.0851i)15-s + (−3.40 − 1.49i)16-s + (0.0665 + 0.0332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 503 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(503\)
Sign: $0.999 + 0.0212i$
Analytic conductor: \(4.01647\)
Root analytic conductor: \(2.00411\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{503} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 503,\ (\ :1/2),\ 0.999 + 0.0212i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627896 - 0.00667168i\)
\(L(\frac12)\) \(\approx\) \(0.627896 - 0.00667168i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad503 \( 1 + (-21.0 - 7.79i)T \)
good2 \( 1 + (1.72 + 2.12i)T + (-0.410 + 1.95i)T^{2} \)
3 \( 1 + (-1.69 + 1.47i)T + (0.430 - 2.96i)T^{2} \)
5 \( 1 + (0.0846 + 0.267i)T + (-4.08 + 2.87i)T^{2} \)
7 \( 1 + (1.43 - 4.15i)T + (-5.51 - 4.31i)T^{2} \)
11 \( 1 + (3.38 - 1.09i)T + (8.91 - 6.44i)T^{2} \)
13 \( 1 + (4.38 - 1.47i)T + (10.3 - 7.87i)T^{2} \)
17 \( 1 + (-0.274 - 0.137i)T + (10.2 + 13.5i)T^{2} \)
19 \( 1 + (-3.22 - 7.22i)T + (-12.7 + 14.1i)T^{2} \)
23 \( 1 + (-1.27 + 4.41i)T + (-19.4 - 12.2i)T^{2} \)
29 \( 1 + (0.386 - 3.23i)T + (-28.1 - 6.83i)T^{2} \)
31 \( 1 + (-0.503 - 6.17i)T + (-30.5 + 5.02i)T^{2} \)
37 \( 1 + (0.424 + 0.536i)T + (-8.49 + 36.0i)T^{2} \)
41 \( 1 + (0.732 - 3.72i)T + (-37.9 - 15.5i)T^{2} \)
43 \( 1 + (-0.198 + 1.07i)T + (-40.1 - 15.2i)T^{2} \)
47 \( 1 + (4.90 - 0.184i)T + (46.8 - 3.52i)T^{2} \)
53 \( 1 + (-0.453 + 1.01i)T + (-35.4 - 39.4i)T^{2} \)
59 \( 1 + (3.91 - 9.40i)T + (-41.5 - 41.8i)T^{2} \)
61 \( 1 + (6.49 + 10.7i)T + (-28.2 + 54.0i)T^{2} \)
67 \( 1 + (-3.68 + 2.26i)T + (30.3 - 59.7i)T^{2} \)
71 \( 1 + (1.99 + 12.6i)T + (-67.5 + 21.8i)T^{2} \)
73 \( 1 + (-4.83 - 4.62i)T + (3.19 + 72.9i)T^{2} \)
79 \( 1 + (-0.576 - 10.2i)T + (-78.4 + 8.88i)T^{2} \)
83 \( 1 + (-2.41 - 4.60i)T + (-47.3 + 68.1i)T^{2} \)
89 \( 1 + (-0.228 - 0.869i)T + (-77.5 + 43.6i)T^{2} \)
97 \( 1 + (3.23 + 13.7i)T + (-86.7 + 43.3i)T^{2} \)
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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

−10.73398235688276890604744119342, −9.858357186671961304716605020609, −9.198710307400268295919893044501, −8.388522745664829322601854616267, −7.905493206618178466985500778952, −6.87653656983063786656748617258, −4.99393315036398902481356014360, −3.14674211413861283104552341311, −2.60004840133679859129453021743, −1.73912548738113906491584285934, 0.48501066068131562221525850167, 2.95671752607893659351619952598, 4.48875394994030167914999331922, 5.37779020612273625299784473322, 6.86290589366744169756977398044, 7.45792381978078106187579773476, 8.075140545528531651356009677928, 9.275000133354580884571102981170, 9.672114794976294133756106024437, 10.35841580279411228265638266044

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