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Properties

Label 2-287-7.2-c1-0-23
Degree $2$
Conductor $287$
Sign $-0.808 + 0.588i$
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  + (0.564 − 0.977i)2-s + (−1.46 − 2.53i)3-s + (0.363 + 0.629i)4-s + (1.46 − 2.53i)5-s − 3.30·6-s + (1.22 − 2.34i)7-s + 3.07·8-s + (−2.79 + 4.83i)9-s + (−1.65 − 2.86i)10-s + (1.90 + 3.29i)11-s + (1.06 − 1.84i)12-s − 2.92·13-s + (−1.60 − 2.52i)14-s − 8.58·15-s + (1.00 − 1.74i)16-s + (−1.11 − 1.93i)17-s + ⋯
L(s)  = 1  + (0.398 − 0.691i)2-s + (−0.845 − 1.46i)3-s + (0.181 + 0.314i)4-s + (0.655 − 1.13i)5-s − 1.34·6-s + (0.463 − 0.886i)7-s + 1.08·8-s + (−0.930 + 1.61i)9-s + (−0.522 − 0.905i)10-s + (0.573 + 0.992i)11-s + (0.307 − 0.532i)12-s − 0.812·13-s + (−0.427 − 0.673i)14-s − 2.21·15-s + (0.252 − 0.437i)16-s + (−0.270 − 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.808 + 0.588i$
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: \((2,\ 287,\ (\ :1/2),\ -0.808 + 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.464120 - 1.42667i\)
\(L(\frac12)\) \(\approx\) \(0.464120 - 1.42667i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.22 + 2.34i)T \)
41 \( 1 + T \)
good2 \( 1 + (-0.564 + 0.977i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.46 + 2.53i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.46 + 2.53i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.90 - 3.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.92T + 13T^{2} \)
17 \( 1 + (1.11 + 1.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.06 - 5.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.527 - 0.914i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.63T + 29T^{2} \)
31 \( 1 + (-3.27 - 5.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.67 + 8.10i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 + (0.886 - 1.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.467 - 0.808i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.65 - 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.17 + 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.23 + 9.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.53T + 71T^{2} \)
73 \( 1 + (-5.88 - 10.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.53 + 13.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.86T + 83T^{2} \)
89 \( 1 + (8.11 - 14.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.70T + 97T^{2} \)
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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

−11.94422892891421199603068641982, −10.83236389609506493703135042635, −9.833896586085245660626186340144, −8.273552710807227644644624125996, −7.39686764283557706597284311489, −6.62322507387169255609270927179, −5.19973169902818465742891444959, −4.32846428742201685593973427285, −2.08591924021796490576646812447, −1.27443961252480879036797547552, 2.64021840685113120661501660660, 4.36556414199632414482172147239, 5.27366544135329703725571919520, 6.15773232922907269033783935682, 6.65842725893331944158213025115, 8.479923423456931413207423583056, 9.701976595468972844121419821248, 10.35208339051167713933594892326, 11.19673761495924225934844945489, 11.65556392849255856242756051637

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