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Properties

Label 2-403-13.9-c1-0-2
Degree $2$
Conductor $403$
Sign $-0.904 - 0.426i$
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  + (−0.944 + 1.63i)2-s + (0.990 − 1.71i)3-s + (−0.785 − 1.36i)4-s − 1.89·5-s + (1.87 + 3.24i)6-s + (−0.456 − 0.790i)7-s − 0.810·8-s + (−0.463 − 0.803i)9-s + (1.78 − 3.09i)10-s + (−2.50 + 4.33i)11-s − 3.11·12-s + (−2.31 + 2.76i)13-s + 1.72·14-s + (−1.87 + 3.24i)15-s + (2.33 − 4.04i)16-s + (2.80 + 4.85i)17-s + ⋯
L(s)  = 1  + (−0.668 + 1.15i)2-s + (0.572 − 0.990i)3-s + (−0.392 − 0.680i)4-s − 0.846·5-s + (0.764 + 1.32i)6-s + (−0.172 − 0.298i)7-s − 0.286·8-s + (−0.154 − 0.267i)9-s + (0.565 − 0.979i)10-s + (−0.753 + 1.30i)11-s − 0.898·12-s + (−0.640 + 0.767i)13-s + 0.461·14-s + (−0.484 + 0.839i)15-s + (0.584 − 1.01i)16-s + (0.679 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.904 - 0.426i$
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: \((2,\ 403,\ (\ :1/2),\ -0.904 - 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126475 + 0.565244i\)
\(L(\frac12)\) \(\approx\) \(0.126475 + 0.565244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.31 - 2.76i)T \)
31 \( 1 - T \)
good2 \( 1 + (0.944 - 1.63i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.990 + 1.71i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.89T + 5T^{2} \)
7 \( 1 + (0.456 + 0.790i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.50 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.80 - 4.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.51 - 2.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.32 - 2.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.45 - 4.25i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (2.22 - 3.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.95 + 8.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.24 + 7.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.40T + 47T^{2} \)
53 \( 1 + 7.22T + 53T^{2} \)
59 \( 1 + (0.903 + 1.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.83 + 4.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.96 - 5.13i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.39 + 4.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.42T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + (8.55 - 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.49 - 9.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.11024512708570759723641562526, −10.43025194855660475442332517100, −9.561504852970637038244570542112, −8.422970434398255425890363751457, −7.65776099237524889123184426128, −7.40579334201085451263445957027, −6.52671219781048958232430472563, −5.12404526501576629707713392587, −3.60573709555826222009330819522, −1.91531890532964631764449727179, 0.42463852407736336154190414900, 2.91903582243151643122760583515, 3.17209923592810565594562892838, 4.59237986362543589906335058067, 5.88336378963904392983988212930, 7.70467645527115076522723589465, 8.392278636564679357014777870087, 9.401784473067819374880916606360, 9.877122478738154862924037968529, 10.82836509283438937758460406483

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