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Properties

Label 2-287-41.32-c1-0-16
Degree $2$
Conductor $287$
Sign $-0.0209 + 0.999i$
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  − 0.710i·2-s + (0.708 − 0.708i)3-s + 1.49·4-s − 3.05i·5-s + (−0.503 − 0.503i)6-s + (0.707 − 0.707i)7-s − 2.48i·8-s + 1.99i·9-s − 2.17·10-s + (−0.816 + 0.816i)11-s + (1.05 − 1.05i)12-s + (−4.51 + 4.51i)13-s + (−0.502 − 0.502i)14-s + (−2.16 − 2.16i)15-s + 1.22·16-s + (0.540 + 0.540i)17-s + ⋯
L(s)  = 1  − 0.502i·2-s + (0.408 − 0.408i)3-s + 0.747·4-s − 1.36i·5-s + (−0.205 − 0.205i)6-s + (0.267 − 0.267i)7-s − 0.878i·8-s + 0.665i·9-s − 0.686·10-s + (−0.246 + 0.246i)11-s + (0.305 − 0.305i)12-s + (−1.25 + 1.25i)13-s + (−0.134 − 0.134i)14-s + (−0.558 − 0.558i)15-s + 0.306·16-s + (0.131 + 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0209 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0209 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.0209 + 0.999i$
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: \((2,\ 287,\ (\ :1/2),\ -0.0209 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18954 - 1.21470i\)
\(L(\frac12)\) \(\approx\) \(1.18954 - 1.21470i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.89 + 5.71i)T \)
good2 \( 1 + 0.710iT - 2T^{2} \)
3 \( 1 + (-0.708 + 0.708i)T - 3iT^{2} \)
5 \( 1 + 3.05iT - 5T^{2} \)
11 \( 1 + (0.816 - 0.816i)T - 11iT^{2} \)
13 \( 1 + (4.51 - 4.51i)T - 13iT^{2} \)
17 \( 1 + (-0.540 - 0.540i)T + 17iT^{2} \)
19 \( 1 + (0.910 + 0.910i)T + 19iT^{2} \)
23 \( 1 - 7.35T + 23T^{2} \)
29 \( 1 + (0.167 - 0.167i)T - 29iT^{2} \)
31 \( 1 + 7.78T + 31T^{2} \)
37 \( 1 - 4.77T + 37T^{2} \)
43 \( 1 - 3.37iT - 43T^{2} \)
47 \( 1 + (-0.556 - 0.556i)T + 47iT^{2} \)
53 \( 1 + (3.52 - 3.52i)T - 53iT^{2} \)
59 \( 1 - 1.06T + 59T^{2} \)
61 \( 1 + 2.75iT - 61T^{2} \)
67 \( 1 + (-4.63 - 4.63i)T + 67iT^{2} \)
71 \( 1 + (9.33 - 9.33i)T - 71iT^{2} \)
73 \( 1 - 3.22iT - 73T^{2} \)
79 \( 1 + (6.84 - 6.84i)T - 79iT^{2} \)
83 \( 1 - 4.02T + 83T^{2} \)
89 \( 1 + (-4.95 + 4.95i)T - 89iT^{2} \)
97 \( 1 + (-11.3 - 11.3i)T + 97iT^{2} \)
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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

−11.62623099775970834842075704022, −10.82477625299869950470113492057, −9.626231192078250792411920680729, −8.809728209720843277532123464484, −7.60853160747265840239539172436, −6.99413179453684589704677146985, −5.28216013419998537882148550744, −4.33844261226075251210690649436, −2.51608499062512128567595163678, −1.46208280268260828464838401509, 2.60174657987277316869847028169, 3.26308907252394436070829958448, 5.19001621482032805434922733828, 6.27162975523276620307767744617, 7.22412587703244990794645108614, 7.910565642039059023813569567637, 9.232390392381397902286548850342, 10.34372055298835932178842217661, 10.93741491015879183543712503288, 11.89650247266627499667226888057

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