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Properties

Label 2-287-41.18-c1-0-14
Degree $2$
Conductor $287$
Sign $-0.0650 + 0.997i$
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  + (−0.149 − 0.459i)2-s − 1.52·3-s + (1.42 − 1.03i)4-s + (1.42 − 1.03i)5-s + (0.228 + 0.702i)6-s + (0.309 − 0.951i)7-s + (−1.47 − 1.07i)8-s − 0.665·9-s + (−0.687 − 0.499i)10-s + (3.16 + 2.30i)11-s + (−2.18 + 1.58i)12-s + (−0.536 − 1.65i)13-s − 0.483·14-s + (−2.17 + 1.57i)15-s + (0.819 − 2.52i)16-s + (−4.22 − 3.07i)17-s + ⋯
L(s)  = 1  + (−0.105 − 0.325i)2-s − 0.882·3-s + (0.714 − 0.519i)4-s + (0.636 − 0.462i)5-s + (0.0931 + 0.286i)6-s + (0.116 − 0.359i)7-s + (−0.520 − 0.378i)8-s − 0.221·9-s + (−0.217 − 0.158i)10-s + (0.955 + 0.694i)11-s + (−0.630 + 0.457i)12-s + (−0.148 − 0.457i)13-s − 0.129·14-s + (−0.561 + 0.407i)15-s + (0.204 − 0.630i)16-s + (−1.02 − 0.744i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0650 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0650 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.0650 + 0.997i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.0650 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770678 - 0.822593i\)
\(L(\frac12)\) \(\approx\) \(0.770678 - 0.822593i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.68 - 2.94i)T \)
good2 \( 1 + (0.149 + 0.459i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + 1.52T + 3T^{2} \)
5 \( 1 + (-1.42 + 1.03i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-3.16 - 2.30i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.536 + 1.65i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.22 + 3.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.68 + 5.18i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.871 - 2.68i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.49 + 3.99i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (6.85 + 4.98i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.216 + 0.157i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-1.94 - 5.97i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.41 - 7.41i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.33 - 4.60i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.61 - 11.1i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.50 - 7.70i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-9.77 + 7.10i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-12.0 - 8.77i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 9.13T + 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 + 3.31T + 83T^{2} \)
89 \( 1 + (1.05 - 3.23i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.42 + 2.48i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.33827392247257203412127258025, −10.97688263568618446536072407830, −9.663140675489424857890930494902, −9.206091547007377117633564884101, −7.38605466206224587826048129191, −6.48514027352598764952542043895, −5.60758225250364146077322117731, −4.58819376930421154929145445976, −2.57193713354458207618173546344, −1.01267799215840513723917304574, 2.08283109569199810740635212468, 3.59772778662528072082790399343, 5.38256036188001599797923180341, 6.38327632519030520736357613111, 6.68048817852987883349758620720, 8.263166958552702182835041394693, 9.050942516134664998232846185761, 10.54495687541504980835973214810, 11.11145478005338048529047882637, 12.00101091835679231199506524316

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