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Properties

Label 2-287-41.37-c1-0-20
Degree $2$
Conductor $287$
Sign $0.261 + 0.965i$
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  + (1.84 − 1.33i)2-s + 1.18·3-s + (0.981 − 3.01i)4-s + (0.156 − 0.482i)5-s + (2.17 − 1.58i)6-s + (−0.809 − 0.587i)7-s + (−0.825 − 2.54i)8-s − 1.60·9-s + (−0.356 − 1.09i)10-s + (1.23 + 3.79i)11-s + (1.16 − 3.57i)12-s + (−2.30 + 1.67i)13-s − 2.27·14-s + (0.185 − 0.570i)15-s + (0.218 + 0.158i)16-s + (−0.771 − 2.37i)17-s + ⋯
L(s)  = 1  + (1.30 − 0.945i)2-s + 0.682·3-s + (0.490 − 1.50i)4-s + (0.0701 − 0.215i)5-s + (0.888 − 0.645i)6-s + (−0.305 − 0.222i)7-s + (−0.292 − 0.898i)8-s − 0.533·9-s + (−0.112 − 0.347i)10-s + (0.371 + 1.14i)11-s + (0.334 − 1.03i)12-s + (−0.639 + 0.464i)13-s − 0.607·14-s + (0.0479 − 0.147i)15-s + (0.0545 + 0.0396i)16-s + (−0.187 − 0.576i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.261 + 0.965i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.261 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22219 - 1.70103i\)
\(L(\frac12)\) \(\approx\) \(2.22219 - 1.70103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (5.11 + 3.85i)T \)
good2 \( 1 + (-1.84 + 1.33i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 - 1.18T + 3T^{2} \)
5 \( 1 + (-0.156 + 0.482i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (-1.23 - 3.79i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.30 - 1.67i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.771 + 2.37i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.643 + 0.467i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.716 - 0.520i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.07 + 6.38i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.287 - 0.885i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.454 + 1.39i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-0.816 + 0.593i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (4.13 - 3.00i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.94 - 12.1i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.55 + 1.85i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.57 + 4.05i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.22 + 6.83i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.79 + 5.53i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 - 0.603T + 73T^{2} \)
79 \( 1 + 1.80T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + (2.28 + 1.65i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.08 - 6.43i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.92715715214336206412657874162, −10.96340467043733160918711452154, −9.835156190014130568571340286964, −9.090971290687301310509332248078, −7.67318430395416560268433176706, −6.44411620049895669673185100976, −5.08272112543780808745556710574, −4.22979749875375814601747823190, −3.03116760417180668193171813463, −2.00853127982973200882429823854, 2.84092954248896719542226111963, 3.61782099775087521828218557928, 5.04221754649161392127408513250, 6.03339278372676972826435701337, 6.80894871098857827847126272905, 8.085671222987848158099196946614, 8.732375086307018608517406109143, 10.11110346971006519715045133506, 11.37174614580407771057936570635, 12.40023006939107292439722590340

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