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Properties

Label 2-287-287.9-c1-0-7
Degree $2$
Conductor $287$
Sign $0.894 + 0.447i$
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  + (−0.866 − 0.5i)2-s + (0.133 − 0.5i)3-s + (−0.500 − 0.866i)4-s + (1.5 + 0.866i)5-s + (−0.366 + 0.366i)6-s + (0.866 + 2.5i)7-s + 3i·8-s + (2.36 + 1.36i)9-s + (−0.866 − 1.5i)10-s + (0.767 − 2.86i)11-s + (−0.5 + 0.133i)12-s + (1.73 + 1.73i)13-s + (0.500 − 2.59i)14-s + (0.633 − 0.633i)15-s + (0.500 − 0.866i)16-s + (1.73 + 0.464i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.0773 − 0.288i)3-s + (−0.250 − 0.433i)4-s + (0.670 + 0.387i)5-s + (−0.149 + 0.149i)6-s + (0.327 + 0.944i)7-s + 1.06i·8-s + (0.788 + 0.455i)9-s + (−0.273 − 0.474i)10-s + (0.231 − 0.864i)11-s + (−0.144 + 0.0386i)12-s + (0.480 + 0.480i)13-s + (0.133 − 0.694i)14-s + (0.163 − 0.163i)15-s + (0.125 − 0.216i)16-s + (0.420 + 0.112i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06815 - 0.252423i\)
\(L(\frac12)\) \(\approx\) \(1.06815 - 0.252423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.866 - 2.5i)T \)
41 \( 1 + (-4 + 5i)T \)
good2 \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.767 + 2.86i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \)
17 \( 1 + (-1.73 - 0.464i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.76 + 6.59i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.267 - 0.464i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.26 - 3.26i)T + 29iT^{2} \)
31 \( 1 + (-1.36 - 2.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.73 - 6.46i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 1.46iT - 43T^{2} \)
47 \( 1 + (-0.901 - 3.36i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.366 + 1.36i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.63 + 6.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.06 + 3.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.36 + 0.633i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.36 - 8.36i)T + 71iT^{2} \)
73 \( 1 + (9.92 - 5.73i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.23 - 1.66i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + 2.73T + 83T^{2} \)
89 \( 1 + (6.83 - 1.83i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.46 + 1.46i)T - 97iT^{2} \)
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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

−11.45217625096400185494455542234, −10.75352911350954388065530661353, −9.860730733556581244535738830917, −8.918354690348926897816657768836, −8.297785863251373342049891887519, −6.76500008719241045490478124299, −5.80302319362105232340094582513, −4.72072251367491895200347085883, −2.64562444613397587515496299948, −1.48576128404845344671969559089, 1.34167477071554674246718186435, 3.71413599463580481172870451494, 4.50704302931421637097617459640, 6.09843035766230369238065281809, 7.28262283224837087863056159854, 7.988325147124534279652760178723, 9.137536526160206734458579170149, 9.930856822237028490184931135166, 10.43091715608613135343532244536, 12.08106164491836350278066456148

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