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Properties

Label 2-287-41.10-c1-0-0
Degree $2$
Conductor $287$
Sign $-0.846 - 0.532i$
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  − 2.23·3-s + (−0.618 − 1.90i)4-s + (0.309 + 0.951i)5-s + (0.809 − 0.587i)7-s + 2.00·9-s + (−1.11 + 3.44i)11-s + (1.38 + 4.25i)12-s + (−5.04 − 3.66i)13-s + (−0.690 − 2.12i)15-s + (−3.23 + 2.35i)16-s + (−1.88 + 5.79i)17-s + (−6.04 + 4.39i)19-s + (1.61 − 1.17i)20-s + (−1.80 + 1.31i)21-s + (0.118 + 0.0857i)23-s + ⋯
L(s)  = 1  − 1.29·3-s + (−0.309 − 0.951i)4-s + (0.138 + 0.425i)5-s + (0.305 − 0.222i)7-s + 0.666·9-s + (−0.337 + 1.03i)11-s + (0.398 + 1.22i)12-s + (−1.39 − 1.01i)13-s + (−0.178 − 0.549i)15-s + (−0.809 + 0.587i)16-s + (−0.456 + 1.40i)17-s + (−1.38 + 1.00i)19-s + (0.361 − 0.262i)20-s + (−0.394 + 0.286i)21-s + (0.0246 + 0.0178i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.846 - 0.532i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.846 - 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0255109 + 0.0884684i\)
\(L(\frac12)\) \(\approx\) \(0.0255109 + 0.0884684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 6.01i)T \)
good2 \( 1 + (0.618 + 1.90i)T^{2} \)
3 \( 1 + 2.23T + 3T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (1.11 - 3.44i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (5.04 + 3.66i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.88 - 5.79i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.04 - 4.39i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.118 - 0.0857i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.19 + 3.66i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.80 - 5.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.781 + 2.40i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (3.04 + 2.21i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-3.73 - 2.71i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.763 + 2.35i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (9.59 + 6.96i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.11 - 2.26i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.47 + 7.60i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (4.38 - 13.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + 1.76T + 79T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (11.1 - 8.11i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.16 + 12.8i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.39614540150462479270797213866, −10.81631468742309251982073293840, −10.55277439253236959490803858937, −9.884779809892696189206785475017, −8.347862031695501256352547161054, −7.01711488393384363519088720490, −6.11620771461763941210175105239, −5.24265414906733168564543342404, −4.38719675407319583878780639613, −1.99974282907645260136250969811, 0.07624734508162267726931561773, 2.67855151998531586328515079829, 4.60628803576230124066531717286, 5.05763581699274236598609621167, 6.48597084713485371484289146870, 7.38992823983176181300293567570, 8.707234783470234364332046081007, 9.354964705057417478339814740044, 10.89863453245752189973790955927, 11.52218919858770390744003698792

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