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Properties

Label 2-287-41.16-c1-0-14
Degree $2$
Conductor $287$
Sign $0.991 - 0.129i$
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 + (1.61 + 1.17i)4-s + (−0.809 − 0.587i)5-s + (−0.309 − 0.951i)7-s + 2.00·9-s + (1.11 − 0.812i)11-s + (3.61 + 2.62i)12-s + (0.545 − 1.67i)13-s + (−1.80 − 1.31i)15-s + (1.23 + 3.80i)16-s + (−4.11 + 2.99i)17-s + (−0.454 − 1.40i)19-s + (−0.618 − 1.90i)20-s + (−0.690 − 2.12i)21-s + (−2.11 + 6.51i)23-s + ⋯
L(s)  = 1  + 1.29·3-s + (0.809 + 0.587i)4-s + (−0.361 − 0.262i)5-s + (−0.116 − 0.359i)7-s + 0.666·9-s + (0.337 − 0.244i)11-s + (1.04 + 0.758i)12-s + (0.151 − 0.465i)13-s + (−0.467 − 0.339i)15-s + (0.309 + 0.951i)16-s + (−0.998 + 0.725i)17-s + (−0.104 − 0.321i)19-s + (−0.138 − 0.425i)20-s + (−0.150 − 0.464i)21-s + (−0.441 + 1.35i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.991 - 0.129i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01834 + 0.131724i\)
\(L(\frac12)\) \(\approx\) \(2.01834 + 0.131724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (3.30 + 5.48i)T \)
good2 \( 1 + (-1.61 - 1.17i)T^{2} \)
3 \( 1 - 2.23T + 3T^{2} \)
5 \( 1 + (0.809 + 0.587i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (-1.11 + 0.812i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.545 + 1.67i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.11 - 2.99i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.454 + 1.40i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.11 - 6.51i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.30 + 1.67i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.690 - 0.502i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-9.28 - 6.74i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-2.54 + 7.83i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (0.736 - 2.26i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.23 + 3.80i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.59 + 4.89i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.881 + 2.71i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-6.47 - 4.70i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (6.61 - 4.80i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + 2.32T + 73T^{2} \)
79 \( 1 + 6.23T + 79T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (3.33 + 10.2i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.66 - 2.66i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.80150187520291300546714447929, −11.02125292874505600175474643556, −9.823697000037752126588096070513, −8.679966304207036966162357686327, −8.090288058065565087465671664152, −7.25555153212601329551119596861, −6.08561181962945822486297601784, −4.11930577537592518722367611845, −3.31755718748994953505151448806, −2.05775835243454030106773562796, 2.02227895550259129114930214339, 2.97586025273578933713172011842, 4.36810882369573082562899974044, 6.03140978654089467330493000953, 7.01560060896927248906610564447, 7.925754324728750825277420796620, 9.047179766247595012214834087387, 9.685608246667631485577814736890, 10.95810809091937125418766908284, 11.62485657019081161898818624502

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