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Properties

Label 2-287-41.32-c1-0-0
Degree $2$
Conductor $287$
Sign $-0.0482 + 0.998i$
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  + 2.24i·2-s + (−1.79 + 1.79i)3-s − 3.04·4-s + 1.17i·5-s + (−4.03 − 4.03i)6-s + (−0.707 + 0.707i)7-s − 2.34i·8-s − 3.43i·9-s − 2.62·10-s + (−2.44 + 2.44i)11-s + (5.46 − 5.46i)12-s + (4.67 − 4.67i)13-s + (−1.58 − 1.58i)14-s + (−2.09 − 2.09i)15-s − 0.813·16-s + (0.749 + 0.749i)17-s + ⋯
L(s)  = 1  + 1.58i·2-s + (−1.03 + 1.03i)3-s − 1.52·4-s + 0.523i·5-s + (−1.64 − 1.64i)6-s + (−0.267 + 0.267i)7-s − 0.830i·8-s − 1.14i·9-s − 0.831·10-s + (−0.737 + 0.737i)11-s + (1.57 − 1.57i)12-s + (1.29 − 1.29i)13-s + (−0.424 − 0.424i)14-s + (−0.542 − 0.542i)15-s − 0.203·16-s + (0.181 + 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.0482 + 0.998i$
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: \((2,\ 287,\ (\ :1/2),\ -0.0482 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.441430 - 0.463251i\)
\(L(\frac12)\) \(\approx\) \(0.441430 - 0.463251i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.04 - 5.63i)T \)
good2 \( 1 - 2.24iT - 2T^{2} \)
3 \( 1 + (1.79 - 1.79i)T - 3iT^{2} \)
5 \( 1 - 1.17iT - 5T^{2} \)
11 \( 1 + (2.44 - 2.44i)T - 11iT^{2} \)
13 \( 1 + (-4.67 + 4.67i)T - 13iT^{2} \)
17 \( 1 + (-0.749 - 0.749i)T + 17iT^{2} \)
19 \( 1 + (-4.52 - 4.52i)T + 19iT^{2} \)
23 \( 1 + 5.77T + 23T^{2} \)
29 \( 1 + (1.42 - 1.42i)T - 29iT^{2} \)
31 \( 1 - 1.24T + 31T^{2} \)
37 \( 1 + 3.43T + 37T^{2} \)
43 \( 1 - 1.38iT - 43T^{2} \)
47 \( 1 + (5.85 + 5.85i)T + 47iT^{2} \)
53 \( 1 + (7.04 - 7.04i)T - 53iT^{2} \)
59 \( 1 - 6.17T + 59T^{2} \)
61 \( 1 + 1.38iT - 61T^{2} \)
67 \( 1 + (3.26 + 3.26i)T + 67iT^{2} \)
71 \( 1 + (5.01 - 5.01i)T - 71iT^{2} \)
73 \( 1 - 6.15iT - 73T^{2} \)
79 \( 1 + (5.75 - 5.75i)T - 79iT^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + (0.0484 - 0.0484i)T - 89iT^{2} \)
97 \( 1 + (9.56 + 9.56i)T + 97iT^{2} \)
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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

−12.58103168518219057176432379670, −11.38817095184673359486415203117, −10.37252182584577348586386082592, −9.844600243329206838151760423504, −8.422187771031606633828977198624, −7.58269578622135954417822625999, −6.29701254806719344379511312509, −5.69001564672757136828763938954, −4.92282712678538168661941311056, −3.52030149064085968435654165616, 0.56428579520509450016741112550, 1.72946901726695337367994250956, 3.39150529237696436794961543035, 4.78934685802941744600043202801, 6.01666916586281864384188154891, 7.06579737709572914008744921300, 8.492186039477856624496663264543, 9.467637510087555622468119559688, 10.65143781672749555744911122441, 11.37111070589092231956742046661

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