Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.973 - 0.227i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.624i·2-s − 2.78·3-s + 1.60·4-s + 0.619i·5-s − 1.73i·6-s − 2.35i·7-s + 2.25i·8-s + 4.74·9-s − 0.387·10-s − 1.21i·11-s − 4.48·12-s + (−0.818 − 3.51i)13-s + 1.46·14-s − 1.72i·15-s + 1.80·16-s + 4.00·17-s + ⋯
L(s)  = 1  + 0.441i·2-s − 1.60·3-s + 0.804·4-s + 0.277i·5-s − 0.710i·6-s − 0.889i·7-s + 0.797i·8-s + 1.58·9-s − 0.122·10-s − 0.366i·11-s − 1.29·12-s + (−0.227 − 0.973i)13-s + 0.392·14-s − 0.445i·15-s + 0.452·16-s + 0.971·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.973 - 0.227i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 403,\ (\ :1/2),\ 0.973 - 0.227i)$
$L(1)$  $\approx$  $1.02939 + 0.118448i$
$L(\frac12)$  $\approx$  $1.02939 + 0.118448i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{13,\;31\}$, \(F_p(T)\) is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (0.818 + 3.51i)T \)
31 \( 1 + iT \)
good2 \( 1 - 0.624iT - 2T^{2} \)
3 \( 1 + 2.78T + 3T^{2} \)
5 \( 1 - 0.619iT - 5T^{2} \)
7 \( 1 + 2.35iT - 7T^{2} \)
11 \( 1 + 1.21iT - 11T^{2} \)
17 \( 1 - 4.00T + 17T^{2} \)
19 \( 1 - 4.95iT - 19T^{2} \)
23 \( 1 - 2.11T + 23T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
37 \( 1 + 10.5iT - 37T^{2} \)
41 \( 1 - 1.88iT - 41T^{2} \)
43 \( 1 + 5.95T + 43T^{2} \)
47 \( 1 - 12.5iT - 47T^{2} \)
53 \( 1 + 2.74T + 53T^{2} \)
59 \( 1 + 7.46iT - 59T^{2} \)
61 \( 1 + 6.44T + 61T^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 + 2.71iT - 71T^{2} \)
73 \( 1 - 7.81iT - 73T^{2} \)
79 \( 1 - 2.11T + 79T^{2} \)
83 \( 1 + 11.0iT - 83T^{2} \)
89 \( 1 - 2.01iT - 89T^{2} \)
97 \( 1 + 5.43iT - 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.03838996573854816931461848606, −10.67907508008208827811066578985, −9.991497522996439015023447678835, −8.085344865657666820282891771750, −7.33148855872841522727856136879, −6.42669290265586463631551069282, −5.77102182076633398581619746952, −4.82067223632716595571387154736, −3.19548477826759079641255433264, −1.04088348207883971394057690950, 1.22977579475967340979005607855, 2.78897353215983437825668939021, 4.62891457322818943466327643885, 5.40878232832419625715331040984, 6.57203664193675505755697518959, 6.96979991935775283319911472114, 8.591631525059695884520017456164, 9.812659191642896516853697382870, 10.52948373816660407318284917903, 11.44753825190231696880815305521

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