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Properties

Label 2-403-403.129-c1-0-8
Degree $2$
Conductor $403$
Sign $-0.883 - 0.468i$
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  + 1.73i·2-s + (0.5 − 0.866i)3-s − 0.999·4-s + (−1.5 + 0.866i)5-s + (1.49 + 0.866i)6-s + (−1.5 − 0.866i)7-s + 1.73i·8-s + (1 + 1.73i)9-s + (−1.49 − 2.59i)10-s + (−4.5 + 2.59i)11-s + (−0.499 + 0.866i)12-s + (3.5 + 0.866i)13-s + (1.49 − 2.59i)14-s + 1.73i·15-s − 5·16-s + (−1.5 + 2.59i)17-s + ⋯
L(s)  = 1  + 1.22i·2-s + (0.288 − 0.499i)3-s − 0.499·4-s + (−0.670 + 0.387i)5-s + (0.612 + 0.353i)6-s + (−0.566 − 0.327i)7-s + 0.612i·8-s + (0.333 + 0.577i)9-s + (−0.474 − 0.821i)10-s + (−1.35 + 0.783i)11-s + (−0.144 + 0.249i)12-s + (0.970 + 0.240i)13-s + (0.400 − 0.694i)14-s + 0.447i·15-s − 1.25·16-s + (−0.363 + 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.883 - 0.468i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.883 - 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.275619 + 1.10823i\)
\(L(\frac12)\) \(\approx\) \(0.275619 + 1.10823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.5 - 0.866i)T \)
31 \( 1 + (-2 + 5.19i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
37 \( 1 + (-4.5 - 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 0.866i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.5 + 4.33i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.5 - 6.06i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−11.56096736057169804053196458134, −10.75324050158422536874033135067, −9.726239077825228888543700143838, −8.329724611230492089991561265145, −7.67639781775342784113380942666, −7.22944095724605288755749303138, −6.20826797106618959509923242449, −5.12686286162292596336215046841, −3.77113991076228180757092342545, −2.21122299172137249248456917266, 0.71852056467545812564747397142, 2.80780584763186210964710432188, 3.44993905342460745312244779296, 4.54438458427938003988811317873, 5.92483029773220856080478019838, 7.26604476374731263869706879953, 8.469358748881886954546217983490, 9.314718386125546341889393353198, 10.03845417224797447766799417478, 11.04903994723641244914457050017

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