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Properties

Label 2-287-41.32-c1-0-17
Degree $2$
Conductor $287$
Sign $-0.554 + 0.832i$
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  − 1.38i·2-s + (1.24 − 1.24i)3-s + 0.0939·4-s − 1.25i·5-s + (−1.71 − 1.71i)6-s + (−0.707 + 0.707i)7-s − 2.89i·8-s − 0.103i·9-s − 1.72·10-s + (−0.995 + 0.995i)11-s + (0.117 − 0.117i)12-s + (1.16 − 1.16i)13-s + (0.976 + 0.976i)14-s + (−1.56 − 1.56i)15-s − 3.80·16-s + (−0.885 − 0.885i)17-s + ⋯
L(s)  = 1  − 0.976i·2-s + (0.719 − 0.719i)3-s + 0.0469·4-s − 0.560i·5-s + (−0.702 − 0.702i)6-s + (−0.267 + 0.267i)7-s − 1.02i·8-s − 0.0344i·9-s − 0.546·10-s + (−0.300 + 0.300i)11-s + (0.0337 − 0.0337i)12-s + (0.323 − 0.323i)13-s + (0.260 + 0.260i)14-s + (−0.402 − 0.402i)15-s − 0.950·16-s + (−0.214 − 0.214i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.805899 - 1.50456i\)
\(L(\frac12)\) \(\approx\) \(0.805899 - 1.50456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (5.50 - 3.26i)T \)
good2 \( 1 + 1.38iT - 2T^{2} \)
3 \( 1 + (-1.24 + 1.24i)T - 3iT^{2} \)
5 \( 1 + 1.25iT - 5T^{2} \)
11 \( 1 + (0.995 - 0.995i)T - 11iT^{2} \)
13 \( 1 + (-1.16 + 1.16i)T - 13iT^{2} \)
17 \( 1 + (0.885 + 0.885i)T + 17iT^{2} \)
19 \( 1 + (-4.74 - 4.74i)T + 19iT^{2} \)
23 \( 1 + 5.26T + 23T^{2} \)
29 \( 1 + (0.807 - 0.807i)T - 29iT^{2} \)
31 \( 1 - 0.629T + 31T^{2} \)
37 \( 1 + 4.95T + 37T^{2} \)
43 \( 1 - 0.969iT - 43T^{2} \)
47 \( 1 + (-0.979 - 0.979i)T + 47iT^{2} \)
53 \( 1 + (-7.56 + 7.56i)T - 53iT^{2} \)
59 \( 1 - 8.26T + 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 + (-1.91 - 1.91i)T + 67iT^{2} \)
71 \( 1 + (-5.24 + 5.24i)T - 71iT^{2} \)
73 \( 1 + 2.81iT - 73T^{2} \)
79 \( 1 + (1.50 - 1.50i)T - 79iT^{2} \)
83 \( 1 - 6.52T + 83T^{2} \)
89 \( 1 + (4.71 - 4.71i)T - 89iT^{2} \)
97 \( 1 + (3.61 + 3.61i)T + 97iT^{2} \)
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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

−11.76930802898751201476043855692, −10.52623626598092480184065856719, −9.764459661431856344183517414776, −8.651995243000662150562898012687, −7.76386257740505258741800947240, −6.74264290094877331209198507763, −5.32298793474161476093274648272, −3.68420313128303771322687478699, −2.55891803179350876880901051188, −1.42059476983288443548038991531, 2.63487747249637089658163689991, 3.77131479788715226498833653283, 5.19710460502273706250039251902, 6.43333075243736808025885854079, 7.17566573881498489201969582775, 8.289201172132828923398250151963, 9.093682130524075395539958000288, 10.15832802828815900071133328472, 11.03553536364489114734018325140, 12.01474540434735652545267297802

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