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Properties

Label 2-299-13.10-c1-0-13
Degree $2$
Conductor $299$
Sign $0.156 + 0.987i$
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  + (−0.385 + 0.222i)2-s + (−0.707 − 1.22i)3-s + (−0.901 + 1.56i)4-s + 2.23i·5-s + (0.544 + 0.314i)6-s + (−4.08 − 2.35i)7-s − 1.69i·8-s + (0.498 − 0.863i)9-s + (−0.496 − 0.860i)10-s + (5.41 − 3.12i)11-s + 2.55·12-s + (1.25 − 3.38i)13-s + 2.09·14-s + (2.73 − 1.58i)15-s + (−1.42 − 2.47i)16-s + (−0.193 + 0.335i)17-s + ⋯
L(s)  = 1  + (−0.272 + 0.157i)2-s + (−0.408 − 0.707i)3-s + (−0.450 + 0.780i)4-s + 0.999i·5-s + (0.222 + 0.128i)6-s + (−1.54 − 0.890i)7-s − 0.597i·8-s + (0.166 − 0.287i)9-s + (−0.157 − 0.272i)10-s + (1.63 − 0.943i)11-s + 0.736·12-s + (0.346 − 0.937i)13-s + 0.560·14-s + (0.707 − 0.408i)15-s + (−0.356 − 0.617i)16-s + (−0.0470 + 0.0814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $0.156 + 0.987i$
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: \((2,\ 299,\ (\ :1/2),\ 0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.485888 - 0.415161i\)
\(L(\frac12)\) \(\approx\) \(0.485888 - 0.415161i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.25 + 3.38i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.385 - 0.222i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.23iT - 5T^{2} \)
7 \( 1 + (4.08 + 2.35i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.41 + 3.12i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.193 - 0.335i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0411 + 0.0237i)T + (9.5 + 16.4i)T^{2} \)
29 \( 1 + (2.90 + 5.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.753iT - 31T^{2} \)
37 \( 1 + (6.77 - 3.91i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.65 - 3.84i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.87 + 8.44i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.02iT - 47T^{2} \)
53 \( 1 - 1.14T + 53T^{2} \)
59 \( 1 + (3.41 + 1.97i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.523 + 0.907i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.717 + 0.414i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.28 + 3.05i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.30iT - 73T^{2} \)
79 \( 1 - 16.2T + 79T^{2} \)
83 \( 1 + 9.03iT - 83T^{2} \)
89 \( 1 + (-3.80 + 2.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.92 + 3.42i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.70759651326554120508518195102, −10.52580272667894897640058703276, −9.663632061088795037895708101483, −8.670123861175420565319809005217, −7.35026459371351157413295805458, −6.70201894891899627070748696295, −6.16945017556481680949409051021, −3.71717620185055695409017766088, −3.38463174197990494742268505591, −0.57494573290875849794173219731, 1.67790425291957114424617232086, 3.94047077542312160499769210128, 4.85555227221697797772068988683, 5.86856406873868965160043360326, 6.82277537920916758604407062044, 8.838571959599816094582715810189, 9.326521874358475681052876026908, 9.707713276848899979340594868308, 10.88256810065402836182019782790, 11.99622048429914132889552705599

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