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Properties

Label 2-287-41.9-c1-0-16
Degree $2$
Conductor $287$
Sign $-0.971 + 0.238i$
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  − 1.55i·2-s + (−0.820 − 0.820i)3-s − 0.415·4-s + 1.07i·5-s + (−1.27 + 1.27i)6-s + (−0.707 − 0.707i)7-s − 2.46i·8-s − 1.65i·9-s + 1.66·10-s + (−0.684 − 0.684i)11-s + (0.340 + 0.340i)12-s + (−3.90 − 3.90i)13-s + (−1.09 + 1.09i)14-s + (0.878 − 0.878i)15-s − 4.65·16-s + (0.0855 − 0.0855i)17-s + ⋯
L(s)  = 1  − 1.09i·2-s + (−0.473 − 0.473i)3-s − 0.207·4-s + 0.479i·5-s + (−0.520 + 0.520i)6-s + (−0.267 − 0.267i)7-s − 0.870i·8-s − 0.551i·9-s + 0.526·10-s + (−0.206 − 0.206i)11-s + (0.0984 + 0.0984i)12-s + (−1.08 − 1.08i)13-s + (−0.293 + 0.293i)14-s + (0.226 − 0.226i)15-s − 1.16·16-s + (0.0207 − 0.0207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.971 + 0.238i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.971 + 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120031 - 0.990136i\)
\(L(\frac12)\) \(\approx\) \(0.120031 - 0.990136i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.94 - 4.07i)T \)
good2 \( 1 + 1.55iT - 2T^{2} \)
3 \( 1 + (0.820 + 0.820i)T + 3iT^{2} \)
5 \( 1 - 1.07iT - 5T^{2} \)
11 \( 1 + (0.684 + 0.684i)T + 11iT^{2} \)
13 \( 1 + (3.90 + 3.90i)T + 13iT^{2} \)
17 \( 1 + (-0.0855 + 0.0855i)T - 17iT^{2} \)
19 \( 1 + (2.20 - 2.20i)T - 19iT^{2} \)
23 \( 1 - 7.33T + 23T^{2} \)
29 \( 1 + (-0.934 - 0.934i)T + 29iT^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + 9.35T + 37T^{2} \)
43 \( 1 + 12.4iT - 43T^{2} \)
47 \( 1 + (-4.96 + 4.96i)T - 47iT^{2} \)
53 \( 1 + (-5.23 - 5.23i)T + 53iT^{2} \)
59 \( 1 - 7.62T + 59T^{2} \)
61 \( 1 + 2.05iT - 61T^{2} \)
67 \( 1 + (-5.85 + 5.85i)T - 67iT^{2} \)
71 \( 1 + (-10.4 - 10.4i)T + 71iT^{2} \)
73 \( 1 - 13.7iT - 73T^{2} \)
79 \( 1 + (4.42 + 4.42i)T + 79iT^{2} \)
83 \( 1 - 7.54T + 83T^{2} \)
89 \( 1 + (0.00682 + 0.00682i)T + 89iT^{2} \)
97 \( 1 + (-0.412 + 0.412i)T - 97iT^{2} \)
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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

−11.44308464724656616055893518008, −10.44363207435735028845149562876, −10.06667189371864469147582339400, −8.694910355557326425474313573849, −7.16262822947178305066938844104, −6.65062537208988690769872095052, −5.22841716772261911131237620007, −3.56212416725491837767768575661, −2.61208930408760292058859776586, −0.78648585493510370566172714974, 2.44798105064458455635342405849, 4.75398569745578246515133987198, 5.04604562643183206039816311269, 6.43795438609602897536606365564, 7.18459253011522244422338261310, 8.352371903655570852874921647017, 9.212594895267636673025999483440, 10.33997104513384601547017916645, 11.33018414212326116165044060360, 12.16290597909756248899000793016

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