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Properties

Label 2-287-41.10-c1-0-17
Degree $2$
Conductor $287$
Sign $0.129 + 0.991i$
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  + (−0.740 − 0.537i)2-s + 3.13·3-s + (−0.359 − 1.10i)4-s + (−0.751 − 2.31i)5-s + (−2.32 − 1.68i)6-s + (0.809 − 0.587i)7-s + (−0.894 + 2.75i)8-s + 6.83·9-s + (−0.687 + 2.11i)10-s + (−0.560 + 1.72i)11-s + (−1.12 − 3.46i)12-s + (−3.08 − 2.23i)13-s − 0.915·14-s + (−2.35 − 7.25i)15-s + (0.261 − 0.190i)16-s + (−0.576 + 1.77i)17-s + ⋯
L(s)  = 1  + (−0.523 − 0.380i)2-s + 1.81·3-s + (−0.179 − 0.552i)4-s + (−0.336 − 1.03i)5-s + (−0.948 − 0.688i)6-s + (0.305 − 0.222i)7-s + (−0.316 + 0.973i)8-s + 2.27·9-s + (−0.217 + 0.669i)10-s + (−0.169 + 0.520i)11-s + (−0.325 − 1.00i)12-s + (−0.854 − 0.620i)13-s − 0.244·14-s + (−0.608 − 1.87i)15-s + (0.0653 − 0.0475i)16-s + (−0.139 + 0.430i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.129 + 0.991i$
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: \((2,\ 287,\ (\ :1/2),\ 0.129 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17576 - 1.03266i\)
\(L(\frac12)\) \(\approx\) \(1.17576 - 1.03266i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.24 - 5.52i)T \)
good2 \( 1 + (0.740 + 0.537i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 - 3.13T + 3T^{2} \)
5 \( 1 + (0.751 + 2.31i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (0.560 - 1.72i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (3.08 + 2.23i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.576 - 1.77i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.69 - 1.23i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.44 - 3.95i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.231 - 0.711i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.75 + 5.40i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.60 - 4.93i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (5.23 + 3.80i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-5.51 - 4.00i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.49 + 10.7i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.93 - 7.21i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.15 + 5.19i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.67 - 14.3i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.23 - 3.79i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 + 3.02T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + (10.9 - 7.95i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.23 + 13.0i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.54424233481127583040238607667, −10.10720016897889106684863801308, −9.713369034047816324560203388008, −8.607624740170286289194616675900, −8.278535745768367105662893608563, −7.21346842444529309033489914983, −5.17167650095240525152172184528, −4.22081662679667597178870354566, −2.64782936980743095247022238844, −1.38561677650250911248884634200, 2.53499827731524333123751731511, 3.27690732079289137064404194848, 4.47730550028515014059765317056, 6.93264094197313051878921612144, 7.24297687075629179678838530010, 8.359680123854797694907775400807, 8.865982414654101705506755562843, 9.762596170298881983562268374611, 10.86737245219477196806361898473, 12.18553226845115030998761011072

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