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Properties

Label 2-299-13.10-c1-0-1
Degree $2$
Conductor $299$
Sign $-0.776 - 0.630i$
Analytic cond. $2.38752$
Root an. cond. $1.54516$
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  + (−0.112 + 0.0647i)2-s + (0.143 + 0.249i)3-s + (−0.991 + 1.71i)4-s + 2.44i·5-s + (−0.0322 − 0.0186i)6-s + (−1.49 − 0.861i)7-s − 0.515i·8-s + (1.45 − 2.52i)9-s + (−0.158 − 0.274i)10-s + (−5.35 + 3.08i)11-s − 0.570·12-s + (1.33 + 3.35i)13-s + 0.223·14-s + (−0.609 + 0.351i)15-s + (−1.94 − 3.37i)16-s + (−0.600 + 1.04i)17-s + ⋯
L(s)  = 1  + (−0.0793 + 0.0458i)2-s + (0.0830 + 0.143i)3-s + (−0.495 + 0.858i)4-s + 1.09i·5-s + (−0.0131 − 0.00760i)6-s + (−0.563 − 0.325i)7-s − 0.182i·8-s + (0.486 − 0.842i)9-s + (−0.0500 − 0.0867i)10-s + (−1.61 + 0.931i)11-s − 0.164·12-s + (0.369 + 0.929i)13-s + 0.0596·14-s + (−0.157 + 0.0908i)15-s + (−0.487 − 0.844i)16-s + (−0.145 + 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $-0.776 - 0.630i$
Analytic conductor: \(2.38752\)
Root analytic conductor: \(1.54516\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{299} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 299,\ (\ :1/2),\ -0.776 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264731 + 0.746252i\)
\(L(\frac12)\) \(\approx\) \(0.264731 + 0.746252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.33 - 3.35i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.112 - 0.0647i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.143 - 0.249i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.44iT - 5T^{2} \)
7 \( 1 + (1.49 + 0.861i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.35 - 3.08i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.600 - 1.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.24 + 1.87i)T + (9.5 + 16.4i)T^{2} \)
29 \( 1 + (-4.30 - 7.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.2iT - 31T^{2} \)
37 \( 1 + (-2.18 + 1.26i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.93 + 2.84i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.259 + 0.448i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.20iT - 47T^{2} \)
53 \( 1 - 1.08T + 53T^{2} \)
59 \( 1 + (-0.990 - 0.571i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.05 + 3.56i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.9 + 6.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.25 - 1.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.71iT - 73T^{2} \)
79 \( 1 - 1.52T + 79T^{2} \)
83 \( 1 - 9.69iT - 83T^{2} \)
89 \( 1 + (1.19 - 0.690i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.65 + 4.99i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.49447186053440751714144351718, −10.96485781466479058153174259911, −10.26138241683114890604096074539, −9.336789358005941790401919159940, −8.313596567831531574550038546316, −6.99880040052626344968400527391, −6.77110877633586505008661684885, −4.79809058153946081066390097382, −3.70334401258093704622290761777, −2.67921971634169489940638566402, 0.58864332777129472387830710599, 2.45939565880330428148719870734, 4.39453932874120025110107391046, 5.40150819602361084397791051559, 6.03704184554156423536862532125, 7.977036606682158459575360244024, 8.376242421610252066192337579621, 9.602947169398094802139998373534, 10.32970343049088745344127059310, 11.17031168164477854299493024534

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