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Properties

Label 2-287-41.16-c1-0-18
Degree $2$
Conductor $287$
Sign $-0.835 + 0.549i$
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  + (0.491 − 1.51i)2-s + 0.558·3-s + (−0.431 − 0.313i)4-s + (−3.26 − 2.36i)5-s + (0.274 − 0.845i)6-s + (−0.309 − 0.951i)7-s + (1.88 − 1.37i)8-s − 2.68·9-s + (−5.18 + 3.77i)10-s + (2.79 − 2.03i)11-s + (−0.241 − 0.175i)12-s + (−0.432 + 1.32i)13-s − 1.59·14-s + (−1.82 − 1.32i)15-s + (−1.47 − 4.54i)16-s + (−0.840 + 0.610i)17-s + ⋯
L(s)  = 1  + (0.347 − 1.07i)2-s + 0.322·3-s + (−0.215 − 0.156i)4-s + (−1.45 − 1.05i)5-s + (0.112 − 0.345i)6-s + (−0.116 − 0.359i)7-s + (0.667 − 0.485i)8-s − 0.895·9-s + (−1.64 + 1.19i)10-s + (0.844 − 0.613i)11-s + (−0.0696 − 0.0505i)12-s + (−0.119 + 0.368i)13-s − 0.425·14-s + (−0.470 − 0.341i)15-s + (−0.369 − 1.13i)16-s + (−0.203 + 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.835 + 0.549i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.384897 - 1.28571i\)
\(L(\frac12)\) \(\approx\) \(0.384897 - 1.28571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-5.36 - 3.48i)T \)
good2 \( 1 + (-0.491 + 1.51i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 - 0.558T + 3T^{2} \)
5 \( 1 + (3.26 + 2.36i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (-2.79 + 2.03i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.432 - 1.32i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.840 - 0.610i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.963 - 2.96i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.74 + 8.43i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-3.01 - 2.19i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-5.95 + 4.32i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.53 - 1.11i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (1.03 - 3.17i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (3.51 - 10.8i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.19 - 1.59i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.02 + 9.31i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.14 + 9.67i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-11.6 - 8.49i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-0.474 + 0.345i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 - 2.73T + 79T^{2} \)
83 \( 1 + 0.190T + 83T^{2} \)
89 \( 1 + (-2.91 - 8.98i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.22 + 3.07i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.47020400142871430622864085488, −11.01834547095645734207835388141, −9.570452360330826672238587235776, −8.523307788807528013002641492968, −7.902479032516954575965738612631, −6.51515212517932071321604399974, −4.67440349882498842538040647116, −3.96345892943955406026252033293, −2.93314573561420148448935113221, −0.926935326409095737467135151385, 2.76995644288765714153388093738, 3.95838567389970301783223350441, 5.28474369677430889716932298728, 6.56421963474840184395382389731, 7.23491746286426584011277898576, 8.012663762121190118444045830559, 9.001178793898908313632579687702, 10.46545108233409759413105768408, 11.53889648557798312973373852618, 11.85823392558645344932440799007

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