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Properties

Label 2-287-41.18-c1-0-5
Degree $2$
Conductor $287$
Sign $0.0400 - 0.999i$
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  + (0.286 + 0.880i)2-s − 2.35·3-s + (0.924 − 0.671i)4-s + (−0.331 + 0.240i)5-s + (−0.674 − 2.07i)6-s + (−0.309 + 0.951i)7-s + (2.35 + 1.71i)8-s + 2.56·9-s + (−0.306 − 0.222i)10-s + (1.68 + 1.22i)11-s + (−2.18 + 1.58i)12-s + (1.56 + 4.82i)13-s − 0.925·14-s + (0.781 − 0.567i)15-s + (−0.126 + 0.388i)16-s + (1.06 + 0.772i)17-s + ⋯
L(s)  = 1  + (0.202 + 0.622i)2-s − 1.36·3-s + (0.462 − 0.335i)4-s + (−0.148 + 0.107i)5-s + (−0.275 − 0.847i)6-s + (−0.116 + 0.359i)7-s + (0.832 + 0.604i)8-s + 0.854·9-s + (−0.0970 − 0.0705i)10-s + (0.507 + 0.368i)11-s + (−0.629 + 0.457i)12-s + (0.434 + 1.33i)13-s − 0.247·14-s + (0.201 − 0.146i)15-s + (−0.0315 + 0.0972i)16-s + (0.257 + 0.187i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0400 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0400 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.0400 - 0.999i$
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: \((2,\ 287,\ (\ :1/2),\ 0.0400 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.743344 + 0.714163i\)
\(L(\frac12)\) \(\approx\) \(0.743344 + 0.714163i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.75 - 2.80i)T \)
good2 \( 1 + (-0.286 - 0.880i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + 2.35T + 3T^{2} \)
5 \( 1 + (0.331 - 0.240i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-1.68 - 1.22i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.56 - 4.82i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.06 - 0.772i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.39 - 4.28i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.00474 - 0.0145i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.06 + 3.67i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (7.40 + 5.37i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.31 - 0.953i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-0.0580 - 0.178i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.229 + 0.705i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.43 + 3.22i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.64 + 11.2i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.51 + 10.8i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.24 - 6.71i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (3.58 + 2.60i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 7.00T + 73T^{2} \)
79 \( 1 + 2.67T + 79T^{2} \)
83 \( 1 - 6.28T + 83T^{2} \)
89 \( 1 + (-0.651 + 2.00i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (6.00 - 4.36i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.72011769058638028725537014949, −11.37400899447891940025574074860, −10.40043021666044510323183816980, −9.328936977363135737908402705303, −7.85622323704354913405483516839, −6.71167142113833905664276576332, −6.18158468252309717647395036922, −5.31321938471206933530313240981, −4.11853659463257841347339156688, −1.75032018383703850910113043629, 0.923611422002601918046921549395, 3.01546874089686315756523266898, 4.29118120924830063859193162501, 5.53335977618148151747717951843, 6.55230556854863170814899339395, 7.45935791664315797102151089051, 8.768806494580583876435824783145, 10.46877694259772463463580529990, 10.66217833023256910774999918151, 11.60420428231682242506844418155

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