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Properties

Label 2-287-7.4-c1-0-10
Degree $2$
Conductor $287$
Sign $-0.483 - 0.875i$
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  + (1.20 + 2.08i)2-s + (0.448 − 0.776i)3-s + (−1.90 + 3.29i)4-s + (−0.448 − 0.776i)5-s + 2.15·6-s + (0.117 + 2.64i)7-s − 4.34·8-s + (1.09 + 1.90i)9-s + (1.07 − 1.87i)10-s + (−0.652 + 1.13i)11-s + (1.70 + 2.95i)12-s + 0.896·13-s + (−5.37 + 3.42i)14-s − 0.803·15-s + (−1.42 − 2.47i)16-s + (1.32 − 2.30i)17-s + ⋯
L(s)  = 1  + (0.851 + 1.47i)2-s + (0.258 − 0.448i)3-s + (−0.950 + 1.64i)4-s + (−0.200 − 0.347i)5-s + 0.881·6-s + (0.0442 + 0.999i)7-s − 1.53·8-s + (0.366 + 0.634i)9-s + (0.341 − 0.591i)10-s + (−0.196 + 0.340i)11-s + (0.492 + 0.852i)12-s + 0.248·13-s + (−1.43 + 0.916i)14-s − 0.207·15-s + (−0.356 − 0.618i)16-s + (0.322 − 0.558i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.483 - 0.875i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.483 - 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01095 + 1.71307i\)
\(L(\frac12)\) \(\approx\) \(1.01095 + 1.71307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.117 - 2.64i)T \)
41 \( 1 + T \)
good2 \( 1 + (-1.20 - 2.08i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.448 + 0.776i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.448 + 0.776i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.652 - 1.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.896T + 13T^{2} \)
17 \( 1 + (-1.32 + 2.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.50 + 2.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.67 + 4.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.155T + 29T^{2} \)
31 \( 1 + (-4.25 + 7.37i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.03 - 1.78i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 4.43T + 43T^{2} \)
47 \( 1 + (1.12 + 1.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.03 + 8.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.91 - 5.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.33 - 2.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.10 - 3.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.45T + 71T^{2} \)
73 \( 1 + (7.00 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.12 - 12.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 + (2.15 + 3.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−12.57753142271455949635771884951, −11.64205882109706027467180944778, −10.06127202525298375949692455664, −8.606014418868341535232957751991, −8.166320436004057169019419173800, −7.13042313393216246534999659793, −6.22640214306964713274233262222, −5.13201653354661508014941887059, −4.35337305251035312706967162320, −2.52428238375729260954924751300, 1.37888708210959600307973133890, 3.28278340169382457419401472611, 3.76854643482262177768681129187, 4.84540649761199154129730572470, 6.27339494256712566746177615080, 7.66567997385056122722216736290, 9.088947229481930122664343534941, 10.22185946381664717433584412366, 10.50516683065059351108983160631, 11.51063805116190537977317921474

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