Properties

Label 2-287-287.286-c2-0-6
Degree $2$
Conductor $287$
Sign $-0.868 + 0.495i$
Analytic cond. $7.82018$
Root an. cond. $2.79645$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.60·2-s − 0.377·3-s + 9.02·4-s + 7.18i·5-s + 1.36·6-s + (−2.11 + 6.67i)7-s − 18.1·8-s − 8.85·9-s − 25.9i·10-s + 13.5i·11-s − 3.40·12-s − 11.7·13-s + (7.63 − 24.0i)14-s − 2.71i·15-s + 29.2·16-s + 31.2·17-s + ⋯
L(s)  = 1  − 1.80·2-s − 0.125·3-s + 2.25·4-s + 1.43i·5-s + 0.226·6-s + (−0.302 + 0.953i)7-s − 2.26·8-s − 0.984·9-s − 2.59i·10-s + 1.23i·11-s − 0.283·12-s − 0.902·13-s + (0.545 − 1.71i)14-s − 0.180i·15-s + 1.83·16-s + 1.83·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.868 + 0.495i$
Analytic conductor: \(7.82018\)
Root analytic conductor: \(2.79645\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1),\ -0.868 + 0.495i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0654201 - 0.246677i\)
\(L(\frac12)\) \(\approx\) \(0.0654201 - 0.246677i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.11 - 6.67i)T \)
41 \( 1 + (30.1 + 27.8i)T \)
good2 \( 1 + 3.60T + 4T^{2} \)
3 \( 1 + 0.377T + 9T^{2} \)
5 \( 1 - 7.18iT - 25T^{2} \)
11 \( 1 - 13.5iT - 121T^{2} \)
13 \( 1 + 11.7T + 169T^{2} \)
17 \( 1 - 31.2T + 289T^{2} \)
19 \( 1 + 15.4T + 361T^{2} \)
23 \( 1 - 12.7T + 529T^{2} \)
29 \( 1 - 8.50iT - 841T^{2} \)
31 \( 1 + 39.3iT - 961T^{2} \)
37 \( 1 + 5.11T + 1.36e3T^{2} \)
43 \( 1 - 21.8T + 1.84e3T^{2} \)
47 \( 1 + 34.3T + 2.20e3T^{2} \)
53 \( 1 - 41.0iT - 2.80e3T^{2} \)
59 \( 1 + 27.2iT - 3.48e3T^{2} \)
61 \( 1 + 77.6iT - 3.72e3T^{2} \)
67 \( 1 - 87.5iT - 4.48e3T^{2} \)
71 \( 1 - 60.1iT - 5.04e3T^{2} \)
73 \( 1 - 89.0iT - 5.32e3T^{2} \)
79 \( 1 + 56.8iT - 6.24e3T^{2} \)
83 \( 1 - 52.4iT - 6.88e3T^{2} \)
89 \( 1 + 78.5T + 7.92e3T^{2} \)
97 \( 1 - 71.6T + 9.40e3T^{2} \)
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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

−11.77090731689142835630292624624, −10.93057721478546561848730711087, −10.00856504192237199934989049953, −9.548089878721802541017734236946, −8.349801043603268944042013308941, −7.43195120713117881739734243054, −6.68223868704083110169248461226, −5.61404533245448194126783748817, −2.98823434449902836842721681139, −2.19820482489438954945705488344, 0.25093628460059494823831519566, 1.19323722053878786388543344834, 3.17194647665717706608847181805, 5.14142596867133745734974667780, 6.32300936073553550493898307467, 7.63347683070870396430073898214, 8.348464959493183218629120873524, 9.018989735646580011447203207935, 9.954055171047391993872459668301, 10.74659727190823023438932759985

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