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Properties

Label 2-287-41.40-c1-0-0
Degree $2$
Conductor $287$
Sign $-0.0630 + 0.998i$
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  − 1.44·2-s + 3.10i·3-s + 0.0984·4-s − 3.65·5-s − 4.50i·6-s + i·7-s + 2.75·8-s − 6.66·9-s + 5.29·10-s − 3.49i·11-s + 0.306i·12-s + 5.95i·13-s − 1.44i·14-s − 11.3i·15-s − 4.18·16-s − 1.79i·17-s + ⋯
L(s)  = 1  − 1.02·2-s + 1.79i·3-s + 0.0492·4-s − 1.63·5-s − 1.83i·6-s + 0.377i·7-s + 0.973·8-s − 2.22·9-s + 1.67·10-s − 1.05i·11-s + 0.0883i·12-s + 1.65i·13-s − 0.387i·14-s − 2.93i·15-s − 1.04·16-s − 0.434i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0630 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0630 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.0630 + 0.998i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.0630 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0391150 - 0.0416660i\)
\(L(\frac12)\) \(\approx\) \(0.0391150 - 0.0416660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
41 \( 1 + (6.39 + 0.404i)T \)
good2 \( 1 + 1.44T + 2T^{2} \)
3 \( 1 - 3.10iT - 3T^{2} \)
5 \( 1 + 3.65T + 5T^{2} \)
11 \( 1 + 3.49iT - 11T^{2} \)
13 \( 1 - 5.95iT - 13T^{2} \)
17 \( 1 + 1.79iT - 17T^{2} \)
19 \( 1 + 2.37iT - 19T^{2} \)
23 \( 1 - 5.32T + 23T^{2} \)
29 \( 1 + 4.93iT - 29T^{2} \)
31 \( 1 + 0.651T + 31T^{2} \)
37 \( 1 + 9.06T + 37T^{2} \)
43 \( 1 + 7.41T + 43T^{2} \)
47 \( 1 - 2.96iT - 47T^{2} \)
53 \( 1 + 4.25iT - 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 4.80T + 61T^{2} \)
67 \( 1 - 4.03iT - 67T^{2} \)
71 \( 1 - 8.10iT - 71T^{2} \)
73 \( 1 - 0.0731T + 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + 9.42T + 83T^{2} \)
89 \( 1 - 5.09iT - 89T^{2} \)
97 \( 1 - 6.72iT - 97T^{2} \)
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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

−11.73147666757920378545683830065, −11.36759767481556031207153457772, −10.59812895241755237784424088934, −9.466832581627686571304911754709, −8.821904221341984218836727177997, −8.309148566941191632044713004824, −6.94329391937374161746730337932, −5.03744237087756188838747681776, −4.27609708286718203543363990741, −3.31688770921343737159556408975, 0.06579849058282776868923995970, 1.40105426968644163854362106121, 3.35854959699572536014760803448, 5.04616671760431864288378210621, 6.90653076559076040136430212518, 7.42800983961654633785032032129, 8.084388897156941987907734393435, 8.666536639532327222258292767275, 10.30071545601191475363905488292, 11.11608133705014820194099840248

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