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Properties

Label 2-287-41.37-c1-0-5
Degree $2$
Conductor $287$
Sign $-0.943 - 0.331i$
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  + (−2.17 + 1.58i)2-s + 2.12·3-s + (1.62 − 4.98i)4-s + (−1.11 + 3.42i)5-s + (−4.63 + 3.36i)6-s + (−0.809 − 0.587i)7-s + (2.69 + 8.30i)8-s + 1.53·9-s + (−2.99 − 9.23i)10-s + (−0.168 − 0.518i)11-s + (3.45 − 10.6i)12-s + (−3.69 + 2.68i)13-s + 2.69·14-s + (−2.37 + 7.30i)15-s + (−10.5 − 7.64i)16-s + (1.03 + 3.17i)17-s + ⋯
L(s)  = 1  + (−1.53 + 1.11i)2-s + 1.22·3-s + (0.810 − 2.49i)4-s + (−0.498 + 1.53i)5-s + (−1.89 + 1.37i)6-s + (−0.305 − 0.222i)7-s + (0.953 + 2.93i)8-s + 0.511·9-s + (−0.948 − 2.91i)10-s + (−0.0508 − 0.156i)11-s + (0.996 − 3.06i)12-s + (−1.02 + 0.743i)13-s + 0.719·14-s + (−0.612 + 1.88i)15-s + (−2.63 − 1.91i)16-s + (0.249 + 0.769i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.943 - 0.331i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114172 + 0.669583i\)
\(L(\frac12)\) \(\approx\) \(0.114172 + 0.669583i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.07 + 2.02i)T \)
good2 \( 1 + (2.17 - 1.58i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 - 2.12T + 3T^{2} \)
5 \( 1 + (1.11 - 3.42i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (0.168 + 0.518i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.69 - 2.68i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.03 - 3.17i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-6.61 - 4.80i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (4.25 - 3.08i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.970 + 2.98i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.624 - 1.92i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.86 + 5.74i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-0.352 + 0.256i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (7.00 - 5.09i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.17 + 3.62i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.11 + 2.98i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.60 - 1.16i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.354 - 1.09i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.301 - 0.928i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 - 8.64T + 73T^{2} \)
79 \( 1 - 16.4T + 79T^{2} \)
83 \( 1 - 2.77T + 83T^{2} \)
89 \( 1 + (-6.17 - 4.48i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.41 - 4.36i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.84444029004205412032826526887, −10.80234387395759871055177873096, −9.834066598488114183241994781660, −9.465925926933024141308349202781, −8.028901171890373742750967691835, −7.70334685591310082935607570550, −6.89023200513849975661881396333, −5.84925203648388952569221460704, −3.57823237389507119604672669178, −2.18044777255226686362679436212, 0.71430991689391196582538365868, 2.42057550442822235851031753751, 3.35613252792071095995008793752, 4.85078065637891503556995807685, 7.39592388416921066455327711734, 7.995780361173549012589712224997, 8.736648674054589198574748321602, 9.470232841927070873986707419994, 9.902484804181356287827697618246, 11.47055677373660824247246004662

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