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Properties

Label 2-287-41.9-c1-0-3
Degree $2$
Conductor $287$
Sign $0.669 - 0.743i$
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  + 0.865i·2-s + (−1.36 − 1.36i)3-s + 1.25·4-s + 1.87i·5-s + (1.18 − 1.18i)6-s + (−0.707 − 0.707i)7-s + 2.81i·8-s + 0.728i·9-s − 1.62·10-s + (3.45 + 3.45i)11-s + (−1.70 − 1.70i)12-s + (1.70 + 1.70i)13-s + (0.611 − 0.611i)14-s + (2.56 − 2.56i)15-s + 0.0693·16-s + (5.13 − 5.13i)17-s + ⋯
L(s)  = 1  + 0.611i·2-s + (−0.788 − 0.788i)3-s + 0.625·4-s + 0.838i·5-s + (0.482 − 0.482i)6-s + (−0.267 − 0.267i)7-s + 0.994i·8-s + 0.242i·9-s − 0.513·10-s + (1.04 + 1.04i)11-s + (−0.493 − 0.493i)12-s + (0.473 + 0.473i)13-s + (0.163 − 0.163i)14-s + (0.661 − 0.661i)15-s + 0.0173·16-s + (1.24 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.669 - 0.743i$
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: \((2,\ 287,\ (\ :1/2),\ 0.669 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12929 + 0.502897i\)
\(L(\frac12)\) \(\approx\) \(1.12929 + 0.502897i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 + 6.14i)T \)
good2 \( 1 - 0.865iT - 2T^{2} \)
3 \( 1 + (1.36 + 1.36i)T + 3iT^{2} \)
5 \( 1 - 1.87iT - 5T^{2} \)
11 \( 1 + (-3.45 - 3.45i)T + 11iT^{2} \)
13 \( 1 + (-1.70 - 1.70i)T + 13iT^{2} \)
17 \( 1 + (-5.13 + 5.13i)T - 17iT^{2} \)
19 \( 1 + (2.15 - 2.15i)T - 19iT^{2} \)
23 \( 1 - 0.329T + 23T^{2} \)
29 \( 1 + (-6.15 - 6.15i)T + 29iT^{2} \)
31 \( 1 + 6.34T + 31T^{2} \)
37 \( 1 + 7.52T + 37T^{2} \)
43 \( 1 + 7.83iT - 43T^{2} \)
47 \( 1 + (-2.19 + 2.19i)T - 47iT^{2} \)
53 \( 1 + (7.63 + 7.63i)T + 53iT^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 + 1.26iT - 61T^{2} \)
67 \( 1 + (-1.96 + 1.96i)T - 67iT^{2} \)
71 \( 1 + (3.10 + 3.10i)T + 71iT^{2} \)
73 \( 1 + 7.35iT - 73T^{2} \)
79 \( 1 + (-3.55 - 3.55i)T + 79iT^{2} \)
83 \( 1 + 17.5T + 83T^{2} \)
89 \( 1 + (-9.35 - 9.35i)T + 89iT^{2} \)
97 \( 1 + (1.37 - 1.37i)T - 97iT^{2} \)
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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

−12.07783897477163358538325290310, −11.13803332863509564445913454429, −10.29385782604444570820762141116, −9.008159686517567782730282176038, −7.41764282980842685680638749073, −6.94162227570265980520862992796, −6.42072904481700538616843423558, −5.27136164616797929536363159561, −3.41826551075252745592002233992, −1.67719264419545094044220654360, 1.20885265156103960434473773594, 3.25041498390663125691833920198, 4.33703444509574412607784351578, 5.74364076046628508435809666979, 6.31286469293340265108971312574, 8.038807724565586380336395461641, 9.073440871817831184856006907980, 10.11760086329241003277900498750, 10.84065858438163176834346487318, 11.55637558011679386459504754965

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