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Properties

Label 2-403-13.3-c1-0-10
Degree $2$
Conductor $403$
Sign $-0.682 + 0.731i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
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  + (−1.21 − 2.11i)2-s + (0.165 + 0.287i)3-s + (−1.97 + 3.41i)4-s − 0.0487·5-s + (0.403 − 0.699i)6-s + (0.449 − 0.779i)7-s + 4.72·8-s + (1.44 − 2.50i)9-s + (0.0594 + 0.102i)10-s + (0.844 + 1.46i)11-s − 1.30·12-s + (3.17 + 1.71i)13-s − 2.19·14-s + (−0.00807 − 0.0139i)15-s + (−1.82 − 3.15i)16-s + (0.475 − 0.823i)17-s + ⋯
L(s)  = 1  + (−0.861 − 1.49i)2-s + (0.0956 + 0.165i)3-s + (−0.985 + 1.70i)4-s − 0.0217·5-s + (0.164 − 0.285i)6-s + (0.170 − 0.294i)7-s + 1.67·8-s + (0.481 − 0.834i)9-s + (0.0187 + 0.0325i)10-s + (0.254 + 0.440i)11-s − 0.377·12-s + (0.879 + 0.475i)13-s − 0.586·14-s + (−0.00208 − 0.00361i)15-s + (−0.455 − 0.789i)16-s + (0.115 − 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.682 + 0.731i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.682 + 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.352736 - 0.811404i\)
\(L(\frac12)\) \(\approx\) \(0.352736 - 0.811404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.17 - 1.71i)T \)
31 \( 1 + T \)
good2 \( 1 + (1.21 + 2.11i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.165 - 0.287i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.0487T + 5T^{2} \)
7 \( 1 + (-0.449 + 0.779i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.844 - 1.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.475 + 0.823i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.28 + 5.69i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.33 + 5.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.14 + 7.17i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-3.18 - 5.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.617 - 1.07i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.26 + 2.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.70T + 47T^{2} \)
53 \( 1 + 7.18T + 53T^{2} \)
59 \( 1 + (-3.11 + 5.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.30 + 7.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.69 - 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.34 + 9.26i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.14T + 73T^{2} \)
79 \( 1 + 3.73T + 79T^{2} \)
83 \( 1 + 4.08T + 83T^{2} \)
89 \( 1 + (-1.62 - 2.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.93 - 8.54i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.98910095411226655953366172867, −9.871970171430904030586143181921, −9.474319600310970547200915370064, −8.578915803380355688830471485350, −7.52732719052907397187962870641, −6.31069924459854133503599430839, −4.40203712813975386425680516323, −3.67419920115619048512216699207, −2.29101164751048245916311792107, −0.873193494126859091674888037806, 1.50451522163114032238574634947, 3.78616151859464042731733225086, 5.52398117322540160727482906287, 5.83030348028240600384680548466, 7.24421655742876607519501109000, 7.83326374747039840096097159032, 8.550456791017656915099775850957, 9.518411557168977245911496179480, 10.34447568399179223524863228503, 11.35960992991374007522933979138

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