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Properties

Label 2-287-41.40-c1-0-8
Degree $2$
Conductor $287$
Sign $0.793 - 0.608i$
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  + 0.152·2-s + 1.15i·3-s − 1.97·4-s + 3.83·5-s + 0.175i·6-s − i·7-s − 0.605·8-s + 1.67·9-s + 0.583·10-s − 2.58i·11-s − 2.27i·12-s + 6.41i·13-s − 0.152i·14-s + 4.42i·15-s + 3.86·16-s + 4.32i·17-s + ⋯
L(s)  = 1  + 0.107·2-s + 0.665i·3-s − 0.988·4-s + 1.71·5-s + 0.0715i·6-s − 0.377i·7-s − 0.213·8-s + 0.557·9-s + 0.184·10-s − 0.779i·11-s − 0.657i·12-s + 1.78i·13-s − 0.0406i·14-s + 1.14i·15-s + 0.965·16-s + 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.793 - 0.608i$
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: \((2,\ 287,\ (\ :1/2),\ 0.793 - 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39707 + 0.474337i\)
\(L(\frac12)\) \(\approx\) \(1.39707 + 0.474337i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.89 + 5.07i)T \)
good2 \( 1 - 0.152T + 2T^{2} \)
3 \( 1 - 1.15iT - 3T^{2} \)
5 \( 1 - 3.83T + 5T^{2} \)
11 \( 1 + 2.58iT - 11T^{2} \)
13 \( 1 - 6.41iT - 13T^{2} \)
17 \( 1 - 4.32iT - 17T^{2} \)
19 \( 1 + 1.25iT - 19T^{2} \)
23 \( 1 + 2.31T + 23T^{2} \)
29 \( 1 + 8.41iT - 29T^{2} \)
31 \( 1 - 4.14T + 31T^{2} \)
37 \( 1 + 7.00T + 37T^{2} \)
43 \( 1 + 4.86T + 43T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 - 5.30iT - 53T^{2} \)
59 \( 1 + 2.16T + 59T^{2} \)
61 \( 1 - 4.40T + 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 - 7.94iT - 71T^{2} \)
73 \( 1 + 16.8T + 73T^{2} \)
79 \( 1 + 5.18iT - 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 - 3.97iT - 89T^{2} \)
97 \( 1 - 1.35iT - 97T^{2} \)
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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

−11.97489656592187844986182374420, −10.53398577467240324093935475854, −9.978397764081054176983230875315, −9.262001505416436288093854102791, −8.512007891593208810304373156048, −6.72185903889478091479132705478, −5.79963993328393112502458311286, −4.70097999355632919994301980759, −3.78401893524973307390120185843, −1.77193102457937169064437714846, 1.41801087437488545758264227199, 2.91509700998359645167819904965, 4.89579659665627001516091837616, 5.56713974834395324985895985841, 6.68041100063017821496543168695, 7.893723632963214552071638085821, 8.997428815620583751307972282363, 9.962087403456672355562978223662, 10.27449739342399181032627939567, 12.19451688508692394271073399219

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