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Properties

Label 2-287-41.10-c1-0-16
Degree $2$
Conductor $287$
Sign $0.334 + 0.942i$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
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  + (0.667 + 0.485i)2-s − 0.0680·3-s + (−0.407 − 1.25i)4-s + (−0.581 − 1.79i)5-s + (−0.0454 − 0.0330i)6-s + (0.809 − 0.587i)7-s + (0.846 − 2.60i)8-s − 2.99·9-s + (0.480 − 1.47i)10-s + (0.141 − 0.434i)11-s + (0.0277 + 0.0853i)12-s + (−0.897 − 0.652i)13-s + 0.825·14-s + (0.0395 + 0.121i)15-s + (−0.303 + 0.220i)16-s + (1.37 − 4.21i)17-s + ⋯
L(s)  = 1  + (0.472 + 0.343i)2-s − 0.0392·3-s + (−0.203 − 0.627i)4-s + (−0.260 − 0.800i)5-s + (−0.0185 − 0.0134i)6-s + (0.305 − 0.222i)7-s + (0.299 − 0.921i)8-s − 0.998·9-s + (0.151 − 0.467i)10-s + (0.0425 − 0.130i)11-s + (0.00800 + 0.0246i)12-s + (−0.249 − 0.180i)13-s + 0.220·14-s + (0.0102 + 0.0314i)15-s + (−0.0759 + 0.0552i)16-s + (0.332 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.334 + 0.942i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.334 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10954 - 0.783672i\)
\(L(\frac12)\) \(\approx\) \(1.10954 - 0.783672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (2.00 - 6.07i)T \)
good2 \( 1 + (-0.667 - 0.485i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + 0.0680T + 3T^{2} \)
5 \( 1 + (0.581 + 1.79i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-0.141 + 0.434i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.897 + 0.652i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.37 + 4.21i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.90 + 1.38i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-6.79 - 4.93i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.15 - 6.64i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.00981 - 0.0302i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.150 + 0.462i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-7.17 - 5.21i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (8.71 + 6.33i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.576 - 1.77i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.26 + 5.28i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.13 + 5.91i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.80 + 8.63i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.29 - 7.07i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 2.75T + 73T^{2} \)
79 \( 1 - 7.03T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + (4.80 - 3.48i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.12 - 15.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.63128782140669878185412899413, −10.85013419208286004895133921723, −9.564582379387870829020205375136, −8.876958656496069898912711181961, −7.68741487887911952103426043037, −6.56316861821020689123044570704, −5.16161590357609723076436190316, −4.96104378898793669419162113475, −3.24410680700025764658768674999, −0.951986364003210192199447782654, 2.49983444569640514833384921329, 3.44010999656101662865560346497, 4.69505288744557105406207687169, 5.90862951487207535080862177393, 7.20360880967081190713881947659, 8.185412628636379044716988199895, 8.979922111821966479014819028493, 10.47951821690608380589209082713, 11.23892092430910932805106988281, 11.97025238074886926550463165528

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