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Properties

Label 2-287-287.114-c1-0-5
Degree $2$
Conductor $287$
Sign $0.0759 + 0.997i$
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  + (−1.12 − 0.648i)2-s + (−2.80 − 0.752i)3-s + (−0.159 − 0.276i)4-s + (1.50 + 0.867i)5-s + (2.66 + 2.66i)6-s + (2.55 + 0.701i)7-s + 3.00i·8-s + (4.72 + 2.72i)9-s + (−1.12 − 1.94i)10-s + (−2.25 − 0.603i)11-s + (0.239 + 0.895i)12-s + (3.55 − 3.55i)13-s + (−2.40 − 2.44i)14-s + (−3.56 − 3.56i)15-s + (1.63 − 2.82i)16-s + (−0.348 + 1.30i)17-s + ⋯
L(s)  = 1  + (−0.794 − 0.458i)2-s + (−1.62 − 0.434i)3-s + (−0.0796 − 0.138i)4-s + (0.672 + 0.388i)5-s + (1.08 + 1.08i)6-s + (0.964 + 0.265i)7-s + 1.06i·8-s + (1.57 + 0.909i)9-s + (−0.355 − 0.616i)10-s + (−0.678 − 0.181i)11-s + (0.0692 + 0.258i)12-s + (0.986 − 0.986i)13-s + (−0.644 − 0.652i)14-s + (−0.921 − 0.921i)15-s + (0.407 − 0.706i)16-s + (−0.0846 + 0.315i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0759 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0759 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.0759 + 0.997i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.0759 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408416 - 0.378498i\)
\(L(\frac12)\) \(\approx\) \(0.408416 - 0.378498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.55 - 0.701i)T \)
41 \( 1 + (-5.82 - 2.66i)T \)
good2 \( 1 + (1.12 + 0.648i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (2.80 + 0.752i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-1.50 - 0.867i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.25 + 0.603i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.55 + 3.55i)T - 13iT^{2} \)
17 \( 1 + (0.348 - 1.30i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.18 - 0.586i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.47 + 6.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.651 + 0.651i)T - 29iT^{2} \)
31 \( 1 + (-2.55 - 4.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.37 + 5.84i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 11.9iT - 43T^{2} \)
47 \( 1 + (10.6 - 2.86i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.89 - 2.38i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.182 + 0.316i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.12 - 0.649i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.07 + 15.2i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.32 - 2.32i)T - 71iT^{2} \)
73 \( 1 + (-4.26 + 2.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0273 + 0.101i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 - 6.06T + 83T^{2} \)
89 \( 1 + (0.756 + 2.82i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.441 + 0.441i)T + 97iT^{2} \)
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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

−11.17258386843152108877866452643, −10.67088440375378655835729678909, −10.29643978741932352414188451622, −8.748059006512828286727328378784, −7.87655334627341973677650531441, −6.36860143302980887197583060646, −5.66022214185256078152782654999, −4.84731087612170671764240096643, −2.21636024987814608198650366791, −0.826592484396351056100591201232, 1.22993147283716733873598739326, 4.18006827056145448867011169849, 5.08253608730704470716022611853, 6.11417150358537101459688471509, 7.09570338127414436130127435535, 8.233942121026235854090158272284, 9.348769541206273325691205715721, 10.05996959299780585971352552130, 11.17558774078156213460421546719, 11.60168993681805836671834973998

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