Properties

Label 2-285-15.8-c1-0-15
Degree $2$
Conductor $285$
Sign $0.473 - 0.880i$
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  + (1 + i)2-s + 1.73·3-s + (0.133 + 2.23i)5-s + (1.73 + 1.73i)6-s + (−2.36 + 2.36i)7-s + (2 − 2i)8-s + 2.99·9-s + (−2.09 + 2.36i)10-s − 0.267i·11-s + (−3.73 − 3.73i)13-s − 4.73·14-s + (0.232 + 3.86i)15-s + 4·16-s + (−0.0980 − 0.0980i)17-s + (2.99 + 2.99i)18-s + i·19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00·3-s + (0.0599 + 0.998i)5-s + (0.707 + 0.707i)6-s + (−0.894 + 0.894i)7-s + (0.707 − 0.707i)8-s + 0.999·9-s + (−0.663 + 0.748i)10-s − 0.0807i·11-s + (−1.03 − 1.03i)13-s − 1.26·14-s + (0.0599 + 0.998i)15-s + 16-s + (−0.0237 − 0.0237i)17-s + (0.707 + 0.707i)18-s + 0.229i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.473 - 0.880i$
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: \((2,\ 285,\ (\ :1/2),\ 0.473 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99522 + 1.19216i\)
\(L(\frac12)\) \(\approx\) \(1.99522 + 1.19216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 + (-0.133 - 2.23i)T \)
19 \( 1 - iT \)
good2 \( 1 + (-1 - i)T + 2iT^{2} \)
7 \( 1 + (2.36 - 2.36i)T - 7iT^{2} \)
11 \( 1 + 0.267iT - 11T^{2} \)
13 \( 1 + (3.73 + 3.73i)T + 13iT^{2} \)
17 \( 1 + (0.0980 + 0.0980i)T + 17iT^{2} \)
23 \( 1 + (-6.46 + 6.46i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.73T + 31T^{2} \)
37 \( 1 + (-2.46 + 2.46i)T - 37iT^{2} \)
41 \( 1 - 7.66iT - 41T^{2} \)
43 \( 1 + (-0.0980 - 0.0980i)T + 43iT^{2} \)
47 \( 1 + (3.56 + 3.56i)T + 47iT^{2} \)
53 \( 1 + (2.26 - 2.26i)T - 53iT^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 4.26T + 61T^{2} \)
67 \( 1 + (-8.19 + 8.19i)T - 67iT^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (-3.63 - 3.63i)T + 73iT^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (-3.73 + 3.73i)T - 83iT^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + (-0.464 + 0.464i)T - 97iT^{2} \)
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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

−12.61135665092876412116125467653, −10.83467207164926154177016217365, −9.975519764105539642136165494054, −9.224498254604325294448997757687, −7.83825989190960427555045942117, −6.97989142028256711186109007211, −6.14942158745902905207805660040, −4.97000683161779010420473336791, −3.43857208348695483733203357334, −2.53889979090532287321565885670, 1.78380827156182502251509727929, 3.24042145976041346336922040501, 4.13886816902657200943781201097, 5.04752844673550433200145111900, 7.02656122053206795253881785777, 7.73500448075238860308366174663, 9.113329456238924027485631239416, 9.586374032728513605916289640822, 10.82199011611392517135685350811, 12.00200558759672507992774670073

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