↑

Properties

Label 2-287-287.27-c1-0-10
Degree $2$
Conductor $287$
Sign $0.771 - 0.636i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0425 − 0.0425i)2-s + (0.956 + 2.30i)3-s − 1.99i·4-s + (1.57 + 1.57i)5-s + (0.0576 − 0.139i)6-s + (2.13 − 1.56i)7-s + (−0.170 + 0.170i)8-s + (−2.29 + 2.29i)9-s − 0.133i·10-s + (0.213 + 0.516i)11-s + (4.61 − 1.90i)12-s + (−0.742 − 1.79i)13-s + (−0.157 − 0.0243i)14-s + (−2.12 + 5.13i)15-s − 3.97·16-s + (−0.341 + 0.824i)17-s + ⋯
L(s)  = 1  + (−0.0301 − 0.0301i)2-s + (0.552 + 1.33i)3-s − 0.998i·4-s + (0.703 + 0.703i)5-s + (0.0235 − 0.0567i)6-s + (0.806 − 0.590i)7-s + (−0.0601 + 0.0601i)8-s + (−0.765 + 0.765i)9-s − 0.0423i·10-s + (0.0644 + 0.155i)11-s + (1.33 − 0.551i)12-s + (−0.205 − 0.496i)13-s + (−0.0420 − 0.00650i)14-s + (−0.549 + 1.32i)15-s − 0.994·16-s + (−0.0828 + 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.771 - 0.636i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61459 + 0.580332i\)
\(L(\frac12)\) \(\approx\) \(1.61459 + 0.580332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.13 + 1.56i)T \)
41 \( 1 + (1.98 - 6.08i)T \)
good2 \( 1 + (0.0425 + 0.0425i)T + 2iT^{2} \)
3 \( 1 + (-0.956 - 2.30i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.57 - 1.57i)T + 5iT^{2} \)
11 \( 1 + (-0.213 - 0.516i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (0.742 + 1.79i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + (0.341 - 0.824i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.436 - 1.05i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 - 3.63iT - 23T^{2} \)
29 \( 1 + (3.37 - 1.39i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 8.13T + 31T^{2} \)
37 \( 1 - 7.94T + 37T^{2} \)
43 \( 1 + (5.71 + 5.71i)T + 43iT^{2} \)
47 \( 1 + (-1.70 + 4.12i)T + (-33.2 - 33.2i)T^{2} \)
53 \( 1 + (4.82 + 11.6i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + 6.92iT - 59T^{2} \)
61 \( 1 + (-4.93 + 4.93i)T - 61iT^{2} \)
67 \( 1 + (4.21 + 1.74i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-4.10 + 1.70i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (6.99 - 6.99i)T - 73iT^{2} \)
79 \( 1 + (-0.385 - 0.931i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 - 6.61iT - 83T^{2} \)
89 \( 1 + (3.53 + 8.54i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + (14.4 + 5.99i)T + (68.5 + 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.32772612545349757995081145508, −10.74410892320622552275908163361, −9.975379332116543970699404966203, −9.539191552646609745834619304123, −8.316621710100195166996343207360, −6.96711917068746263589045107553, −5.65484116828719157337453037034, −4.79215210274032731508251574837, −3.57180346321423824292461574618, −1.97624911079241714279222492023, 1.69337380800636249102361593565, 2.68095724072465710413881315389, 4.49382123961919955223313907027, 5.86367821386847535921297991651, 7.10128120063998686385813942361, 7.86475618186459441982873550794, 8.749141995616902076330288931905, 9.262163184845821422184808599992, 11.14415594982994236645698637012, 12.03955916282738146684464566988

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