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Properties

Label 2-287-7.4-c1-0-21
Degree $2$
Conductor $287$
Sign $0.947 + 0.319i$
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  + (0.894 + 1.54i)2-s + (1.16 − 2.00i)3-s + (−0.600 + 1.04i)4-s + (−1.16 − 2.00i)5-s + 4.15·6-s + (−1.87 − 1.87i)7-s + 1.42·8-s + (−1.19 − 2.06i)9-s + (2.07 − 3.59i)10-s + (−1.05 + 1.82i)11-s + (1.39 + 2.41i)12-s + 2.32·13-s + (1.22 − 4.57i)14-s − 5.38·15-s + (2.47 + 4.29i)16-s + (0.622 − 1.07i)17-s + ⋯
L(s)  = 1  + (0.632 + 1.09i)2-s + (0.669 − 1.16i)3-s + (−0.300 + 0.520i)4-s + (−0.518 − 0.898i)5-s + 1.69·6-s + (−0.706 − 0.707i)7-s + 0.504·8-s + (−0.397 − 0.688i)9-s + (0.656 − 1.13i)10-s + (−0.318 + 0.550i)11-s + (0.402 + 0.697i)12-s + 0.643·13-s + (0.328 − 1.22i)14-s − 1.39·15-s + (0.619 + 1.07i)16-s + (0.150 − 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.947 + 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96081 - 0.322084i\)
\(L(\frac12)\) \(\approx\) \(1.96081 - 0.322084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.87 + 1.87i)T \)
41 \( 1 + T \)
good2 \( 1 + (-0.894 - 1.54i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.16 + 2.00i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.16 + 2.00i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.05 - 1.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.32T + 13T^{2} \)
17 \( 1 + (-0.622 + 1.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.21 - 2.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.93 - 5.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + (1.26 - 2.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.200 - 0.347i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 0.872T + 43T^{2} \)
47 \( 1 + (-1.99 - 3.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.03 - 10.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.25 + 9.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.91 + 3.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.54 + 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.22T + 71T^{2} \)
73 \( 1 + (3.22 - 5.58i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.09 + 3.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.83T + 83T^{2} \)
89 \( 1 + (6.26 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−12.41722483580462366795094356012, −10.95629408131257925656443330569, −9.581434609294928136787517957341, −8.397489892255037427867826909577, −7.55859342494460745566167107768, −7.10856800338773593509575316137, −5.96402786365169812030906437719, −4.74189739849726280928449152162, −3.50922053139028650305250851601, −1.40404897157678454948839910104, 2.65008240419087173523914137655, 3.31828554185243684447927770490, 4.06018229922405820352259572088, 5.44059629552965727556502794628, 6.93693611271073261059517709086, 8.341373204793687685072441992856, 9.309089710454543908381312552182, 10.27838309800723905532476310599, 10.93969092039484892215481668563, 11.63248317692979013299075090916

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