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Properties

Label 2-287-41.10-c1-0-2
Degree $2$
Conductor $287$
Sign $0.896 - 0.444i$
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  + (−2.02 − 1.46i)2-s + 0.931·3-s + (1.31 + 4.03i)4-s + (0.525 + 1.61i)5-s + (−1.88 − 1.36i)6-s + (0.809 − 0.587i)7-s + (1.72 − 5.31i)8-s − 2.13·9-s + (1.31 − 4.03i)10-s + (−1.56 + 4.80i)11-s + (1.22 + 3.75i)12-s + (1.62 + 1.17i)13-s − 2.49·14-s + (0.489 + 1.50i)15-s + (−4.44 + 3.22i)16-s + (−1.89 + 5.84i)17-s + ⋯
L(s)  = 1  + (−1.42 − 1.03i)2-s + 0.537·3-s + (0.655 + 2.01i)4-s + (0.235 + 0.723i)5-s + (−0.768 − 0.558i)6-s + (0.305 − 0.222i)7-s + (0.611 − 1.88i)8-s − 0.710·9-s + (0.415 − 1.27i)10-s + (−0.470 + 1.44i)11-s + (0.352 + 1.08i)12-s + (0.449 + 0.326i)13-s − 0.667·14-s + (0.126 + 0.389i)15-s + (−1.11 + 0.807i)16-s + (−0.460 + 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.896 - 0.444i$
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: \((2,\ 287,\ (\ :1/2),\ 0.896 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663042 + 0.155277i\)
\(L(\frac12)\) \(\approx\) \(0.663042 + 0.155277i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.27 - 1.27i)T \)
good2 \( 1 + (2.02 + 1.46i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 - 0.931T + 3T^{2} \)
5 \( 1 + (-0.525 - 1.61i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (1.56 - 4.80i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.62 - 1.17i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.89 - 5.84i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.40 - 1.74i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-3.91 - 2.84i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.884 + 2.72i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.87 + 8.83i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.86 - 5.73i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-7.57 - 5.50i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (2.08 + 1.51i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.84 + 5.66i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.89 - 2.83i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (6.59 - 4.79i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.60 + 8.03i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.58 + 7.95i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 0.703T + 73T^{2} \)
79 \( 1 - 1.29T + 79T^{2} \)
83 \( 1 + 6.20T + 83T^{2} \)
89 \( 1 + (3.64 - 2.65i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.23 - 9.95i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.42479105382321453591897417725, −10.82053118556203241375911862961, −10.00728286919742437648968631813, −9.221390071981118045805093849069, −8.211462463506475901041129727230, −7.58910012476911792145976602040, −6.28586670772560180150061675460, −4.14943884888211371196510557324, −2.75617812322519654259937474360, −1.87169435158794640092582841280, 0.76808171797219545147372739384, 2.78893440157921420460408944175, 5.13041179740849228187702574891, 5.92648603843955214845498244809, 7.15679445662562100585795555732, 8.245723530838756396138821834825, 8.899879169812921866417989419285, 9.103850831778271244614307281963, 10.68996691758552758134867913798, 11.21354317551794035118990251132

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