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Properties

Label 2-299-299.298-c0-0-1
Degree $2$
Conductor $299$
Sign $-0.5 - 0.866i$
Analytic cond. $0.149220$
Root an. cond. $0.386290$
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  + 1.73i·2-s + 3-s − 1.99·4-s + 1.73i·6-s − 1.73i·8-s − 1.99·12-s + (−0.5 − 0.866i)13-s + 0.999·16-s + 23-s − 1.73i·24-s − 25-s + (1.49 − 0.866i)26-s − 27-s + 29-s − 1.73i·31-s + ⋯
L(s)  = 1  + 1.73i·2-s + 3-s − 1.99·4-s + 1.73i·6-s − 1.73i·8-s − 1.99·12-s + (−0.5 − 0.866i)13-s + 0.999·16-s + 23-s − 1.73i·24-s − 25-s + (1.49 − 0.866i)26-s − 27-s + 29-s − 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(0.149220\)
Root analytic conductor: \(0.386290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{299} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 299,\ (\ :0),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8954039990\)
\(L(\frac12)\) \(\approx\) \(0.8954039990\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - T \)
good2 \( 1 - 1.73iT - T^{2} \)
3 \( 1 - T + T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.73iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−12.83949012573380679617956160119, −11.35178310082819962025992300330, −9.811851259047877182381029023785, −9.162775319717803647669518381090, −7.992230075006215209797060145817, −7.83672528861352550132147166782, −6.51701102924783292866005693065, −5.53997564825700863054053837111, −4.38694882469111608925244772706, −2.88721162230758505042251847365, 1.88092230576229406005545877285, 2.93259124862598649941371580953, 3.89296134173483712226722780313, 5.10111466135734875747980559909, 6.99286413255207523000643916783, 8.420761824812220129140559741266, 9.042779541808501970566479858786, 9.853909886785602230718483442971, 10.75022599035494003512053676103, 11.73148891518737979095591072646

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