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Properties

Label 2-285-15.8-c1-0-1
Degree $2$
Conductor $285$
Sign $0.501 - 0.865i$
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  + (−0.631 − 0.631i)2-s + (−1.41 − 0.998i)3-s − 1.20i·4-s + (1.34 + 1.78i)5-s + (0.262 + 1.52i)6-s + (−3.10 + 3.10i)7-s + (−2.02 + 2.02i)8-s + (1.00 + 2.82i)9-s + (0.282 − 1.97i)10-s + 0.484i·11-s + (−1.20 + 1.70i)12-s + (−1.71 − 1.71i)13-s + 3.92·14-s + (−0.111 − 3.87i)15-s + 0.146·16-s + (5.25 + 5.25i)17-s + ⋯
L(s)  = 1  + (−0.446 − 0.446i)2-s + (−0.816 − 0.576i)3-s − 0.601i·4-s + (0.599 + 0.800i)5-s + (0.107 + 0.622i)6-s + (−1.17 + 1.17i)7-s + (−0.714 + 0.714i)8-s + (0.334 + 0.942i)9-s + (0.0893 − 0.624i)10-s + 0.146i·11-s + (−0.346 + 0.491i)12-s + (−0.477 − 0.477i)13-s + 1.04·14-s + (−0.0286 − 0.999i)15-s + 0.0365·16-s + (1.27 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.501 - 0.865i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.501 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420403 + 0.242358i\)
\(L(\frac12)\) \(\approx\) \(0.420403 + 0.242358i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.41 + 0.998i)T \)
5 \( 1 + (-1.34 - 1.78i)T \)
19 \( 1 + iT \)
good2 \( 1 + (0.631 + 0.631i)T + 2iT^{2} \)
7 \( 1 + (3.10 - 3.10i)T - 7iT^{2} \)
11 \( 1 - 0.484iT - 11T^{2} \)
13 \( 1 + (1.71 + 1.71i)T + 13iT^{2} \)
17 \( 1 + (-5.25 - 5.25i)T + 17iT^{2} \)
23 \( 1 + (2.66 - 2.66i)T - 23iT^{2} \)
29 \( 1 + 6.46T + 29T^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 + (1.24 - 1.24i)T - 37iT^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (1.78 + 1.78i)T + 43iT^{2} \)
47 \( 1 + (4.00 + 4.00i)T + 47iT^{2} \)
53 \( 1 + (-2.13 + 2.13i)T - 53iT^{2} \)
59 \( 1 + 4.08T + 59T^{2} \)
61 \( 1 + 0.395T + 61T^{2} \)
67 \( 1 + (0.787 - 0.787i)T - 67iT^{2} \)
71 \( 1 - 7.63iT - 71T^{2} \)
73 \( 1 + (0.661 + 0.661i)T + 73iT^{2} \)
79 \( 1 - 7.50iT - 79T^{2} \)
83 \( 1 + (6.64 - 6.64i)T - 83iT^{2} \)
89 \( 1 - 8.36T + 89T^{2} \)
97 \( 1 + (7.06 - 7.06i)T - 97iT^{2} \)
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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

−11.91502309240686602565656030768, −10.98252029146850936986580455746, −9.919988739492511187326123337801, −9.756040665071593907454458600009, −8.207629592162331473913736453326, −6.79618364957728993214039486231, −5.89001897770066870039403074947, −5.51943890583662148363126469339, −3.00070762468311194225555375563, −1.81165915638047607742526741549, 0.45378417046543989548389836995, 3.38651476526177250284675879592, 4.46649748674798041341521412666, 5.79725070734236338105379425368, 6.77290522526560240168797173703, 7.63475344648381103267336844942, 9.146657246367612973572124200416, 9.672664392116368280409037704889, 10.38527692577490478602178924050, 11.86905838481363158230003094077

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