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Properties

Label 2-403-13.9-c1-0-3
Degree $2$
Conductor $403$
Sign $-0.480 + 0.876i$
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  + (−1.00 + 1.73i)2-s + (−0.398 + 0.691i)3-s + (−1.00 − 1.74i)4-s − 2.51·5-s + (−0.799 − 1.38i)6-s + (2.36 + 4.10i)7-s + 0.0356·8-s + (1.18 + 2.04i)9-s + (2.51 − 4.35i)10-s + (−1.57 + 2.72i)11-s + 1.61·12-s + (2.62 + 2.47i)13-s − 9.49·14-s + (1.00 − 1.73i)15-s + (1.98 − 3.43i)16-s + (−2.08 − 3.60i)17-s + ⋯
L(s)  = 1  + (−0.708 + 1.22i)2-s + (−0.230 + 0.398i)3-s + (−0.504 − 0.873i)4-s − 1.12·5-s + (−0.326 − 0.565i)6-s + (0.895 + 1.55i)7-s + 0.0125·8-s + (0.393 + 0.682i)9-s + (0.796 − 1.37i)10-s + (−0.473 + 0.820i)11-s + 0.464·12-s + (0.727 + 0.686i)13-s − 2.53·14-s + (0.258 − 0.448i)15-s + (0.495 − 0.858i)16-s + (−0.505 − 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.480 + 0.876i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.480 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267265 - 0.451300i\)
\(L(\frac12)\) \(\approx\) \(0.267265 - 0.451300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-2.62 - 2.47i)T \)
31 \( 1 - T \)
good2 \( 1 + (1.00 - 1.73i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.398 - 0.691i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.51T + 5T^{2} \)
7 \( 1 + (-2.36 - 4.10i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.57 - 2.72i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.08 + 3.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.78 + 6.55i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.893 + 1.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.52 + 2.63i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (3.21 - 5.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.81 - 6.60i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.66 - 2.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.68T + 47T^{2} \)
53 \( 1 - 5.60T + 53T^{2} \)
59 \( 1 + (3.47 + 6.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.933 + 1.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.05 + 1.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.16 - 3.74i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 4.81T + 79T^{2} \)
83 \( 1 + 2.04T + 83T^{2} \)
89 \( 1 + (9.32 - 16.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.35 + 4.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.58553878626769466431195534879, −11.12510162868969711753985153960, −9.699192203265107061024926300867, −8.753647616437376105566008722058, −8.253331504884555782779569383109, −7.36050823879638443353707681786, −6.44259564683553836637596755778, −5.07237554759202441344643433933, −4.56861485042823515242806079574, −2.43967677993764959540219777726, 0.46640288489893300662439020477, 1.54548855620681858771019152286, 3.67038059770434354880980008868, 3.95201799101620331947016400365, 5.91706618162577262598419621421, 7.24900203646013366731818304335, 8.119938102738335163201062256894, 8.671962320840044859199686226267, 10.35028511083129509462001543896, 10.63290670892479827856695223260

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