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Properties

Label 2-299-13.10-c1-0-11
Degree $2$
Conductor $299$
Sign $0.986 - 0.161i$
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  + (−2.30 + 1.33i)2-s + (0.800 + 1.38i)3-s + (2.54 − 4.41i)4-s − 0.558i·5-s + (−3.69 − 2.13i)6-s + (−2.36 − 1.36i)7-s + 8.24i·8-s + (0.219 − 0.380i)9-s + (0.744 + 1.28i)10-s + (4.99 − 2.88i)11-s + 8.15·12-s + (−3.38 − 1.23i)13-s + 7.27·14-s + (0.774 − 0.447i)15-s + (−5.88 − 10.2i)16-s + (2.84 − 4.92i)17-s + ⋯
L(s)  = 1  + (−1.63 + 0.941i)2-s + (0.461 + 0.800i)3-s + (1.27 − 2.20i)4-s − 0.249i·5-s + (−1.50 − 0.870i)6-s + (−0.893 − 0.515i)7-s + 2.91i·8-s + (0.0731 − 0.126i)9-s + (0.235 + 0.407i)10-s + (1.50 − 0.868i)11-s + 2.35·12-s + (−0.939 − 0.341i)13-s + 1.94·14-s + (0.199 − 0.115i)15-s + (−1.47 − 2.55i)16-s + (0.688 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(299\)    =    \(13 \cdot 23\)
Sign: $0.986 - 0.161i$
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: \((2,\ 299,\ (\ :1/2),\ 0.986 - 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627576 + 0.0510111i\)
\(L(\frac12)\) \(\approx\) \(0.627576 + 0.0510111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.38 + 1.23i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (2.30 - 1.33i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.800 - 1.38i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.558iT - 5T^{2} \)
7 \( 1 + (2.36 + 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.99 + 2.88i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.84 + 4.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.97 + 2.29i)T + (9.5 + 16.4i)T^{2} \)
29 \( 1 + (-2.65 - 4.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.11iT - 31T^{2} \)
37 \( 1 + (0.473 - 0.273i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.94 - 1.12i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.37 - 4.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 + 2.18T + 53T^{2} \)
59 \( 1 + (6.99 + 4.03i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.74 + 11.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.25 + 4.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.07 - 5.23i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.46iT - 73T^{2} \)
79 \( 1 + 8.71T + 79T^{2} \)
83 \( 1 + 3.95iT - 83T^{2} \)
89 \( 1 + (-6.61 + 3.82i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.0736 - 0.0425i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.26266175083948257550578179756, −10.29189325284857473462416893595, −9.492084273482776334035348793690, −9.176072821460327021006611071497, −8.201354874404308609705164513826, −6.92033695257173827236476206404, −6.41969682754607456197980071151, −4.84841650361660366090085246983, −3.17664787297573562879474688307, −0.793835844923974420493645819626, 1.59683258325389310817619145758, 2.50836132102520889241019227058, 3.86729183169034986219960339076, 6.54679239957036893421481082853, 7.09793543768036388185633796168, 8.189980674968547267795896040921, 8.950445527136966351640829725896, 9.870508932456590259668278515322, 10.41843718067161415035169984693, 11.85753334256594206972090125047

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