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Properties

Label 2-287-41.9-c1-0-7
Degree $2$
Conductor $287$
Sign $0.787 + 0.616i$
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  − 1.08i·2-s + (−0.429 − 0.429i)3-s + 0.818·4-s + 2.95i·5-s + (−0.466 + 0.466i)6-s + (0.707 + 0.707i)7-s − 3.06i·8-s − 2.63i·9-s + 3.20·10-s + (2.06 + 2.06i)11-s + (−0.351 − 0.351i)12-s + (3.10 + 3.10i)13-s + (0.768 − 0.768i)14-s + (1.26 − 1.26i)15-s − 1.69·16-s + (2.42 − 2.42i)17-s + ⋯
L(s)  = 1  − 0.768i·2-s + (−0.248 − 0.248i)3-s + 0.409·4-s + 1.32i·5-s + (−0.190 + 0.190i)6-s + (0.267 + 0.267i)7-s − 1.08i·8-s − 0.876i·9-s + 1.01·10-s + (0.622 + 0.622i)11-s + (−0.101 − 0.101i)12-s + (0.861 + 0.861i)13-s + (0.205 − 0.205i)14-s + (0.327 − 0.327i)15-s − 0.422·16-s + (0.587 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.787 + 0.616i$
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: \((2,\ 287,\ (\ :1/2),\ 0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39210 - 0.480090i\)
\(L(\frac12)\) \(\approx\) \(1.39210 - 0.480090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.25 + 1.37i)T \)
good2 \( 1 + 1.08iT - 2T^{2} \)
3 \( 1 + (0.429 + 0.429i)T + 3iT^{2} \)
5 \( 1 - 2.95iT - 5T^{2} \)
11 \( 1 + (-2.06 - 2.06i)T + 11iT^{2} \)
13 \( 1 + (-3.10 - 3.10i)T + 13iT^{2} \)
17 \( 1 + (-2.42 + 2.42i)T - 17iT^{2} \)
19 \( 1 + (-0.970 + 0.970i)T - 19iT^{2} \)
23 \( 1 + 2.02T + 23T^{2} \)
29 \( 1 + (6.91 + 6.91i)T + 29iT^{2} \)
31 \( 1 - 1.47T + 31T^{2} \)
37 \( 1 + 1.64T + 37T^{2} \)
43 \( 1 - 9.43iT - 43T^{2} \)
47 \( 1 + (7.62 - 7.62i)T - 47iT^{2} \)
53 \( 1 + (-7.23 - 7.23i)T + 53iT^{2} \)
59 \( 1 - 2.37T + 59T^{2} \)
61 \( 1 + 1.14iT - 61T^{2} \)
67 \( 1 + (-4.46 + 4.46i)T - 67iT^{2} \)
71 \( 1 + (2.44 + 2.44i)T + 71iT^{2} \)
73 \( 1 + 12.5iT - 73T^{2} \)
79 \( 1 + (2.72 + 2.72i)T + 79iT^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (7.52 + 7.52i)T + 89iT^{2} \)
97 \( 1 + (-4.94 + 4.94i)T - 97iT^{2} \)
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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

−11.59288280683500802248520155315, −11.12640898004676251775516762151, −9.963041667522263742275978509358, −9.283163279504886333897694553935, −7.56056663518049485423555684981, −6.69671400396949847670972395121, −6.10208536287172215035331292033, −4.02618716402056635890468219803, −3.01479611590892963992417045585, −1.64752638133339674453891503929, 1.54010465381744328900057886086, 3.74355540421592569076023907344, 5.29544046490321113686549633707, 5.61180769752703467027320884744, 7.05196732857831494419805007609, 8.368609559407424788741235037859, 8.444324717230458220549926425075, 10.11346161829649071660262130571, 11.02977950283670276057187587224, 11.82635162123702188008785418904

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