Properties

Degree 2
Conductor 283
Sign $1$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + 1.41i·3-s − 1.00·4-s + 1.41i·5-s + 2.00·6-s − 7-s − 1.00·9-s + 2.00·10-s + 11-s − 1.41i·12-s + 13-s + 1.41i·14-s − 2.00·15-s − 0.999·16-s + 1.41i·18-s − 1.41i·19-s + ⋯
L(s)  = 1  − 1.41i·2-s + 1.41i·3-s − 1.00·4-s + 1.41i·5-s + 2.00·6-s − 7-s − 1.00·9-s + 2.00·10-s + 11-s − 1.41i·12-s + 13-s + 1.41i·14-s − 2.00·15-s − 0.999·16-s + 1.41i·18-s − 1.41i·19-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(283\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{283} (282, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 283,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.7340935769$
$L(\frac12)$  $\approx$  $0.7340935769$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 283$, \(F_p\) is a polynomial of degree 2. If $p = 283$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad283 \( 1 - T \)
good2 \( 1 + 1.41iT - T^{2} \)
3 \( 1 - 1.41iT - T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−11.39136675263032134582558647840, −11.20085922411613741128934633187, −10.28001116438507809233334943749, −9.643362890406966006321985614313, −9.018906507862609046506322422309, −7.00139444185545752330422739223, −6.03694925877457915720953650028, −4.08369552296286370575299516695, −3.62059166261996352957104122593, −2.59467010063561747330397905207, 1.51045195254865270478888566135, 3.98308326875307031073002572314, 5.62659148225833184489551741695, 6.23875193646694958971335889960, 7.05568083941234697100906010314, 8.157220923687053708219498646763, 8.653637518830049935763110848196, 9.693425451728701652986693020310, 11.58219006169165744610692060487, 12.47105597613220777211149229066

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