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Properties

Label 2-287-41.9-c1-0-4
Degree $2$
Conductor $287$
Sign $-0.997 - 0.0640i$
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  + 2.30i·2-s + (1.78 + 1.78i)3-s − 3.29·4-s − 0.414i·5-s + (−4.10 + 4.10i)6-s + (0.707 + 0.707i)7-s − 2.97i·8-s + 3.37i·9-s + 0.954·10-s + (−0.962 − 0.962i)11-s + (−5.87 − 5.87i)12-s + (−0.368 − 0.368i)13-s + (−1.62 + 1.62i)14-s + (0.740 − 0.740i)15-s + 0.255·16-s + (1.81 − 1.81i)17-s + ⋯
L(s)  = 1  + 1.62i·2-s + (1.03 + 1.03i)3-s − 1.64·4-s − 0.185i·5-s + (−1.67 + 1.67i)6-s + (0.267 + 0.267i)7-s − 1.05i·8-s + 1.12i·9-s + 0.301·10-s + (−0.290 − 0.290i)11-s + (−1.69 − 1.69i)12-s + (−0.102 − 0.102i)13-s + (−0.434 + 0.434i)14-s + (0.191 − 0.191i)15-s + 0.0637·16-s + (0.440 − 0.440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.997 - 0.0640i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.997 - 0.0640i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0524712 + 1.63636i\)
\(L(\frac12)\) \(\approx\) \(0.0524712 + 1.63636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-5.93 + 2.39i)T \)
good2 \( 1 - 2.30iT - 2T^{2} \)
3 \( 1 + (-1.78 - 1.78i)T + 3iT^{2} \)
5 \( 1 + 0.414iT - 5T^{2} \)
11 \( 1 + (0.962 + 0.962i)T + 11iT^{2} \)
13 \( 1 + (0.368 + 0.368i)T + 13iT^{2} \)
17 \( 1 + (-1.81 + 1.81i)T - 17iT^{2} \)
19 \( 1 + (0.568 - 0.568i)T - 19iT^{2} \)
23 \( 1 - 0.152T + 23T^{2} \)
29 \( 1 + (0.674 + 0.674i)T + 29iT^{2} \)
31 \( 1 - 6.16T + 31T^{2} \)
37 \( 1 + 0.457T + 37T^{2} \)
43 \( 1 - 0.616iT - 43T^{2} \)
47 \( 1 + (8.26 - 8.26i)T - 47iT^{2} \)
53 \( 1 + (8.08 + 8.08i)T + 53iT^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 - 8.78iT - 61T^{2} \)
67 \( 1 + (-5.61 + 5.61i)T - 67iT^{2} \)
71 \( 1 + (-6.77 - 6.77i)T + 71iT^{2} \)
73 \( 1 + 10.6iT - 73T^{2} \)
79 \( 1 + (-4.59 - 4.59i)T + 79iT^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + (6.56 + 6.56i)T + 89iT^{2} \)
97 \( 1 + (4.30 - 4.30i)T - 97iT^{2} \)
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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

−12.61137229335371049785693933535, −11.11699413777676804191893302328, −9.882305637560074587897856699337, −9.125378962868518929259782410990, −8.340014437642513216084933598762, −7.71194186345074642303865744717, −6.38016319253024467297035037206, −5.18431763499911724715067170451, −4.41190132770527128032866330607, −2.94897914784265318531303403291, 1.31630345373974286668963084502, 2.44098122499927490380159077863, 3.36390399829759746369006579826, 4.71599182305508551929467387797, 6.64729302074982531698444528293, 7.76162739614456276450556039268, 8.615366738138025932799996695939, 9.612649979620591984459792097127, 10.53630836166068436153808137323, 11.41325899895455673279783534258

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