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Properties

Label 2-287-7.2-c1-0-5
Degree $2$
Conductor $287$
Sign $-0.343 - 0.939i$
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  + (−0.985 + 1.70i)2-s + (−0.257 − 0.446i)3-s + (−0.940 − 1.62i)4-s + (0.257 − 0.446i)5-s + 1.01·6-s + (1.91 + 1.82i)7-s − 0.232·8-s + (1.36 − 2.36i)9-s + (0.507 + 0.879i)10-s + (2.24 + 3.88i)11-s + (−0.485 + 0.840i)12-s − 0.515·13-s + (−4.99 + 1.47i)14-s − 0.265·15-s + (2.11 − 3.65i)16-s + (2.69 + 4.66i)17-s + ⋯
L(s)  = 1  + (−0.696 + 1.20i)2-s + (−0.148 − 0.257i)3-s + (−0.470 − 0.814i)4-s + (0.115 − 0.199i)5-s + 0.414·6-s + (0.725 + 0.688i)7-s − 0.0822·8-s + (0.455 − 0.789i)9-s + (0.160 + 0.278i)10-s + (0.676 + 1.17i)11-s + (−0.140 + 0.242i)12-s − 0.142·13-s + (−1.33 + 0.394i)14-s − 0.0686·15-s + (0.527 − 0.914i)16-s + (0.652 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.343 - 0.939i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.343 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534759 + 0.765157i\)
\(L(\frac12)\) \(\approx\) \(0.534759 + 0.765157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.91 - 1.82i)T \)
41 \( 1 + T \)
good2 \( 1 + (0.985 - 1.70i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.257 + 0.446i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.257 + 0.446i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.24 - 3.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.515T + 13T^{2} \)
17 \( 1 + (-2.69 - 4.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.95 - 3.38i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.07 - 1.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.58T + 29T^{2} \)
31 \( 1 + (3.08 + 5.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.76 + 8.25i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 9.50T + 43T^{2} \)
47 \( 1 + (5.13 - 8.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.856 - 1.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.10 + 3.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.70 + 13.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.88 - 3.27i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.37T + 71T^{2} \)
73 \( 1 + (1.74 + 3.02i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.51 + 6.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.25T + 83T^{2} \)
89 \( 1 + (-1.64 + 2.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 19.3T + 97T^{2} \)
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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

−12.41955708891444256858164638347, −11.16833267015962685885363648790, −9.624416306663025871067026938272, −9.260071103394242558500636865427, −8.039985438398302634207596925199, −7.38716502324326484836775077144, −6.28231767793018431395363699282, −5.53629408638069348891096185324, −4.01172856471926842034693949428, −1.67884865473270156429444266859, 1.02332690452219514892186144934, 2.57140395967832264579634936611, 3.94230794677447970302021714193, 5.20941880394734250774664230552, 6.74423471671526591191188940140, 7.995014593129275174189226725879, 8.909245048711236188868045218413, 9.941573223542831902925333828175, 10.69576804117453099764364692984, 11.24286586809300380040595719096

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