Properties

Label 2-283-283.282-c0-0-2
Degree $2$
Conductor $283$
Sign $1$
Analytic cond. $0.141235$
Root an. cond. $0.375812$
Motivic weight $0$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(283\)
Sign: $1$
Analytic conductor: \(0.141235\)
Root analytic conductor: \(0.375812\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (282, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 283,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7340935769\)
\(L(\frac12)\) \(\approx\) \(0.7340935769\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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