Properties

Label 2-287-287.114-c1-0-18
Degree $2$
Conductor $287$
Sign $0.977 - 0.209i$
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 + (1.86 + 0.5i)3-s + (−0.500 − 0.866i)4-s + (1.5 + 0.866i)5-s + (1.36 + 1.36i)6-s + (−0.866 − 2.5i)7-s − 3i·8-s + (0.633 + 0.366i)9-s + (0.866 + 1.5i)10-s + (4.23 + 1.13i)11-s + (−0.500 − 1.86i)12-s + (−1.73 + 1.73i)13-s + (0.500 − 2.59i)14-s + (2.36 + 2.36i)15-s + (0.500 − 0.866i)16-s + (−1.73 + 6.46i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (1.07 + 0.288i)3-s + (−0.250 − 0.433i)4-s + (0.670 + 0.387i)5-s + (0.557 + 0.557i)6-s + (−0.327 − 0.944i)7-s − 1.06i·8-s + (0.211 + 0.122i)9-s + (0.273 + 0.474i)10-s + (1.27 + 0.341i)11-s + (−0.144 − 0.538i)12-s + (−0.480 + 0.480i)13-s + (0.133 − 0.694i)14-s + (0.610 + 0.610i)15-s + (0.125 − 0.216i)16-s + (−0.420 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.977 - 0.209i$
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: \((2,\ 287,\ (\ :1/2),\ 0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29296 + 0.242369i\)
\(L(\frac12)\) \(\approx\) \(2.29296 + 0.242369i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 0.5i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.23 - 1.13i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.73 - 1.73i)T - 13iT^{2} \)
17 \( 1 + (1.73 - 6.46i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.23 - 1.40i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.73 - 6.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.73 + 6.73i)T - 29iT^{2} \)
31 \( 1 + (0.366 + 0.633i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.267 - 0.464i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 5.46iT - 43T^{2} \)
47 \( 1 + (-6.09 + 1.63i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.36 + 0.366i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.36 + 9.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.06 - 3.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.633 - 2.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-6.63 + 6.63i)T - 71iT^{2} \)
73 \( 1 + (-3.92 + 2.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.76 + 10.3i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 - 0.732T + 83T^{2} \)
89 \( 1 + (-1.83 - 6.83i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.46 + 5.46i)T + 97iT^{2} \)
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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

−12.07730815524410339863574896881, −10.53689110021903210671604485883, −9.843553337107640278516672864842, −9.229733533360545749398438380320, −8.007379402534035487431692728077, −6.57294251021861352931270818101, −6.14329860860340135921247432005, −4.22516877690773971038296850296, −3.83916276343148596910061190062, −1.94491572347120926009242755511, 2.27873613165005906994138549951, 2.97870646947108745985572245694, 4.41378305849073306847147518385, 5.57779236430090422374774983672, 6.86403595923064404910108923358, 8.336134923729881378600474461197, 8.869475138896169607852078543565, 9.493408330377044358819768442134, 11.09784283393501639732312597874, 12.23902073518511083763384998711

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