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Properties

Label 2-287-41.40-c1-0-6
Degree $2$
Conductor $287$
Sign $0.874 - 0.485i$
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  − 2.16·2-s + 1.16i·3-s + 2.68·4-s + 2.90·5-s − 2.51i·6-s + i·7-s − 1.47·8-s + 1.64·9-s − 6.29·10-s − 4.29i·11-s + 3.12i·12-s − 1.70i·13-s − 2.16i·14-s + 3.38i·15-s − 2.17·16-s + 4.15i·17-s + ⋯
L(s)  = 1  − 1.52·2-s + 0.671i·3-s + 1.34·4-s + 1.30·5-s − 1.02i·6-s + 0.377i·7-s − 0.521·8-s + 0.548·9-s − 1.99·10-s − 1.29i·11-s + 0.900i·12-s − 0.471i·13-s − 0.578i·14-s + 0.873i·15-s − 0.543·16-s + 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.791327 + 0.204941i\)
\(L(\frac12)\) \(\approx\) \(0.791327 + 0.204941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
41 \( 1 + (-3.10 - 5.59i)T \)
good2 \( 1 + 2.16T + 2T^{2} \)
3 \( 1 - 1.16iT - 3T^{2} \)
5 \( 1 - 2.90T + 5T^{2} \)
11 \( 1 + 4.29iT - 11T^{2} \)
13 \( 1 + 1.70iT - 13T^{2} \)
17 \( 1 - 4.15iT - 17T^{2} \)
19 \( 1 + 5.64iT - 19T^{2} \)
23 \( 1 - 7.57T + 23T^{2} \)
29 \( 1 - 6.78iT - 29T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 - 4.59T + 37T^{2} \)
43 \( 1 + 8.58T + 43T^{2} \)
47 \( 1 + 9.95iT - 47T^{2} \)
53 \( 1 - 7.83iT - 53T^{2} \)
59 \( 1 + 3.09T + 59T^{2} \)
61 \( 1 + 9.99T + 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + 5.04T + 73T^{2} \)
79 \( 1 - 0.438iT - 79T^{2} \)
83 \( 1 + 9.03T + 83T^{2} \)
89 \( 1 - 5.59iT - 89T^{2} \)
97 \( 1 + 9.39iT - 97T^{2} \)
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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

−11.19481328174939263145701801729, −10.66412806473862922982310667563, −9.899716389548214250960655421557, −9.048498953792573245679148848765, −8.616730712655127663671592633869, −7.17310134477662308274727136672, −6.10306785046376139460150622451, −4.95043058344188439207441274142, −2.97255618994297411565536217158, −1.37083408959682063473894719876, 1.35299152607211765619517194177, 2.22870048553023622941484620182, 4.67278226518810598063822499014, 6.29494774209306968279271527161, 7.14855090439090734152155046412, 7.77476858358719762921272028465, 9.193356495564571635646673180214, 9.757118317394285688841395999988, 10.31246614277390384071724367653, 11.51812914549378883945756714752

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