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Properties

Label 2-285-5.4-c1-0-19
Degree $2$
Conductor $285$
Sign $0.0232 - 0.999i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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  − 2.39i·2-s − i·3-s − 3.72·4-s + (−2.23 − 0.0520i)5-s − 2.39·6-s + 4.15i·7-s + 4.13i·8-s − 9-s + (−0.124 + 5.35i)10-s − 5.71·11-s + 3.72i·12-s − 3.79i·13-s + 9.93·14-s + (−0.0520 + 2.23i)15-s + 2.44·16-s − 2.66i·17-s + ⋯
L(s)  = 1  − 1.69i·2-s − 0.577i·3-s − 1.86·4-s + (−0.999 − 0.0232i)5-s − 0.977·6-s + 1.56i·7-s + 1.46i·8-s − 0.333·9-s + (−0.0394 + 1.69i)10-s − 1.72·11-s + 1.07i·12-s − 1.05i·13-s + 2.65·14-s + (−0.0134 + 0.577i)15-s + 0.610·16-s − 0.646i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0232 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0232 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.0232 - 0.999i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.0232 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227525 + 0.222288i\)
\(L(\frac12)\) \(\approx\) \(0.227525 + 0.222288i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (2.23 + 0.0520i)T \)
19 \( 1 + T \)
good2 \( 1 + 2.39iT - 2T^{2} \)
7 \( 1 - 4.15iT - 7T^{2} \)
11 \( 1 + 5.71T + 11T^{2} \)
13 \( 1 + 3.79iT - 13T^{2} \)
17 \( 1 + 2.66iT - 17T^{2} \)
23 \( 1 + 8.13iT - 23T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 + 2.31T + 31T^{2} \)
37 \( 1 + 4.68iT - 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 - 1.76iT - 43T^{2} \)
47 \( 1 - 7.14iT - 47T^{2} \)
53 \( 1 - 3.04iT - 53T^{2} \)
59 \( 1 - 0.582T + 59T^{2} \)
61 \( 1 + 9.54T + 61T^{2} \)
67 \( 1 + 1.20iT - 67T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 + 5.06T + 79T^{2} \)
83 \( 1 + 1.83iT - 83T^{2} \)
89 \( 1 + 3.36T + 89T^{2} \)
97 \( 1 - 0.313iT - 97T^{2} \)
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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

−11.10602818000030434652172629187, −10.66787325405266036922451596892, −9.301912112624473819569352492840, −8.425602513308526550245622889341, −7.62673692645036951933921916346, −5.72588841823063666277236509792, −4.68315433177260459849835956694, −2.97721494964842964018130576974, −2.45267823887578758647189161799, −0.22998919847816911990436653844, 3.74225373175987078109277484190, 4.51108557717169708145020711895, 5.55914981889872827238252482867, 6.99081354487501350376651901247, 7.58245914932968610530888306777, 8.263759582143219399304700025410, 9.479212953004770686657162844778, 10.57436176946308062046396710406, 11.35187849310597494292211983985, 12.99880225168509828178081452281

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