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Properties

Label 2-287-287.286-c2-0-17
Degree $2$
Conductor $287$
Sign $0.993 - 0.114i$
Analytic cond. $7.82018$
Root an. cond. $2.79645$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.517·2-s − 2.41·3-s − 3.73·4-s − 2.97i·5-s − 1.24·6-s + (2.38 + 6.58i)7-s − 4.00·8-s − 3.16·9-s − 1.54i·10-s − 4.27i·11-s + 9.01·12-s + 14.4·13-s + (1.23 + 3.40i)14-s + 7.19i·15-s + 12.8·16-s + 14.2·17-s + ⋯
L(s)  = 1  + 0.258·2-s − 0.804·3-s − 0.933·4-s − 0.595i·5-s − 0.208·6-s + (0.340 + 0.940i)7-s − 0.500·8-s − 0.352·9-s − 0.154i·10-s − 0.388i·11-s + 0.751·12-s + 1.11·13-s + (0.0880 + 0.243i)14-s + 0.479i·15-s + 0.803·16-s + 0.839·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(7.82018\)
Root analytic conductor: \(2.79645\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1),\ 0.993 - 0.114i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08800 + 0.0626920i\)
\(L(\frac12)\) \(\approx\) \(1.08800 + 0.0626920i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.38 - 6.58i)T \)
41 \( 1 + (9.42 - 39.9i)T \)
good2 \( 1 - 0.517T + 4T^{2} \)
3 \( 1 + 2.41T + 9T^{2} \)
5 \( 1 + 2.97iT - 25T^{2} \)
11 \( 1 + 4.27iT - 121T^{2} \)
13 \( 1 - 14.4T + 169T^{2} \)
17 \( 1 - 14.2T + 289T^{2} \)
19 \( 1 - 7.77T + 361T^{2} \)
23 \( 1 + 5.72T + 529T^{2} \)
29 \( 1 - 47.3iT - 841T^{2} \)
31 \( 1 + 47.8iT - 961T^{2} \)
37 \( 1 - 46.7T + 1.36e3T^{2} \)
43 \( 1 - 7.24T + 1.84e3T^{2} \)
47 \( 1 + 1.84T + 2.20e3T^{2} \)
53 \( 1 + 2.40iT - 2.80e3T^{2} \)
59 \( 1 + 57.9iT - 3.48e3T^{2} \)
61 \( 1 - 36.3iT - 3.72e3T^{2} \)
67 \( 1 + 10.0iT - 4.48e3T^{2} \)
71 \( 1 - 64.0iT - 5.04e3T^{2} \)
73 \( 1 + 82.0iT - 5.32e3T^{2} \)
79 \( 1 + 60.4iT - 6.24e3T^{2} \)
83 \( 1 - 112. iT - 6.88e3T^{2} \)
89 \( 1 - 98.9T + 7.92e3T^{2} \)
97 \( 1 - 78.1T + 9.40e3T^{2} \)
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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

−11.71149187039780026732311551787, −10.92234475093889352879320511658, −9.564500821593753590013602285027, −8.734784948440973840884981432884, −8.073465122760309985551710793344, −6.12377549294112221229330453390, −5.54159965647286917683782505582, −4.70628203940776872576869428188, −3.24835002767279074365212619946, −0.951148853863712476483551316792, 0.851149745210916584009570133177, 3.31392093516798656977588090902, 4.41495350225795324473087303875, 5.49535675592755499640429806715, 6.43070828128258358630851512429, 7.66722768533971195021005577761, 8.679813395720695134843942067177, 9.956046259709991403278871762826, 10.67293164173609864253640774653, 11.53792139408249818808382310403

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