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Properties

Label 2-403-403.99-c1-0-6
Degree $2$
Conductor $403$
Sign $0.767 - 0.640i$
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  + (−1.36 − 1.36i)2-s + (0.633 − 0.366i)3-s + 1.73i·4-s + (−1.36 + 0.366i)5-s + (−1.36 − 0.366i)6-s + (−4.23 − 1.13i)7-s + (−0.366 + 0.366i)8-s + (−1.23 + 2.13i)9-s + (2.36 + 1.36i)10-s + (3.23 − 0.866i)11-s + (0.633 + 1.09i)12-s + (3.59 − 0.232i)13-s + (4.23 + 7.33i)14-s + (−0.732 + 0.732i)15-s + 4.46·16-s + ⋯
L(s)  = 1  + (−0.965 − 0.965i)2-s + (0.366 − 0.211i)3-s + 0.866i·4-s + (−0.610 + 0.163i)5-s + (−0.557 − 0.149i)6-s + (−1.59 − 0.428i)7-s + (−0.129 + 0.129i)8-s + (−0.410 + 0.711i)9-s + (0.748 + 0.431i)10-s + (0.974 − 0.261i)11-s + (0.183 + 0.316i)12-s + (0.997 − 0.0643i)13-s + (1.13 + 1.95i)14-s + (−0.189 + 0.189i)15-s + 1.11·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.767 - 0.640i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.767 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371644 + 0.134635i\)
\(L(\frac12)\) \(\approx\) \(0.371644 + 0.134635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.59 + 0.232i)T \)
31 \( 1 + (4.33 - 3.5i)T \)
good2 \( 1 + (1.36 + 1.36i)T + 2iT^{2} \)
3 \( 1 + (-0.633 + 0.366i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.36 - 0.366i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (4.23 + 1.13i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.23 + 0.866i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.96 - 7.33i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 4.19T + 23T^{2} \)
29 \( 1 - 1.53iT - 29T^{2} \)
37 \( 1 + (1.26 - 4.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (8.46 - 2.26i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.09 - 5.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.46 + 8.46i)T - 47iT^{2} \)
53 \( 1 + (9.23 + 5.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.5 + 3.09i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 7.19iT - 61T^{2} \)
67 \( 1 + (-1.13 + 0.303i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.33 + 2.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.83 - 1.56i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.26 - 4.19i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.43 - 12.8i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (5.73 + 5.73i)T + 89iT^{2} \)
97 \( 1 + (1.53 + 1.53i)T + 97iT^{2} \)
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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

−11.12036810397359617327754159459, −10.48205998515865715315670051129, −9.596641017853688321476378915216, −8.762388349070706419952912918139, −8.047021162570699761605908940292, −6.85263212451028725426053849051, −5.82633496234183685250966494809, −3.67699327764048254848188424176, −3.18876973713278519387747870444, −1.49668887256583417189692956503, 0.35109151455652498844391362959, 3.10979634619315297680477703241, 4.00017145345799191075595290064, 5.99432635456497654457165043484, 6.54872542306820916703688463361, 7.40530936352630435532067395391, 8.821949822380357265023216549485, 8.998386248481441708963772876647, 9.650657844184328434287091888547, 10.99991106158652312600851273646

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