Properties

Label 2-285-15.8-c1-0-20
Degree $2$
Conductor $285$
Sign $0.374 + 0.927i$
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  + (−0.538 − 0.538i)2-s + (1.54 − 0.777i)3-s − 1.42i·4-s + (1.01 + 1.99i)5-s + (−1.25 − 0.415i)6-s + (1.88 − 1.88i)7-s + (−1.84 + 1.84i)8-s + (1.79 − 2.40i)9-s + (0.524 − 1.62i)10-s + 1.35i·11-s + (−1.10 − 2.19i)12-s + (1.03 + 1.03i)13-s − 2.02·14-s + (3.12 + 2.29i)15-s − 0.856·16-s + (−1.42 − 1.42i)17-s + ⋯
L(s)  = 1  + (−0.380 − 0.380i)2-s + (0.893 − 0.448i)3-s − 0.710i·4-s + (0.455 + 0.890i)5-s + (−0.511 − 0.169i)6-s + (0.711 − 0.711i)7-s + (−0.651 + 0.651i)8-s + (0.597 − 0.801i)9-s + (0.165 − 0.512i)10-s + 0.409i·11-s + (−0.318 − 0.634i)12-s + (0.288 + 0.288i)13-s − 0.541·14-s + (0.806 + 0.591i)15-s − 0.214·16-s + (−0.345 − 0.345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.374 + 0.927i$
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: \((2,\ 285,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27824 - 0.861901i\)
\(L(\frac12)\) \(\approx\) \(1.27824 - 0.861901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.54 + 0.777i)T \)
5 \( 1 + (-1.01 - 1.99i)T \)
19 \( 1 + iT \)
good2 \( 1 + (0.538 + 0.538i)T + 2iT^{2} \)
7 \( 1 + (-1.88 + 1.88i)T - 7iT^{2} \)
11 \( 1 - 1.35iT - 11T^{2} \)
13 \( 1 + (-1.03 - 1.03i)T + 13iT^{2} \)
17 \( 1 + (1.42 + 1.42i)T + 17iT^{2} \)
23 \( 1 + (1.15 - 1.15i)T - 23iT^{2} \)
29 \( 1 + 4.57T + 29T^{2} \)
31 \( 1 + 4.81T + 31T^{2} \)
37 \( 1 + (-2.99 + 2.99i)T - 37iT^{2} \)
41 \( 1 + 8.00iT - 41T^{2} \)
43 \( 1 + (-5.08 - 5.08i)T + 43iT^{2} \)
47 \( 1 + (-6.64 - 6.64i)T + 47iT^{2} \)
53 \( 1 + (7.34 - 7.34i)T - 53iT^{2} \)
59 \( 1 - 1.44T + 59T^{2} \)
61 \( 1 + 8.86T + 61T^{2} \)
67 \( 1 + (2.63 - 2.63i)T - 67iT^{2} \)
71 \( 1 - 4.15iT - 71T^{2} \)
73 \( 1 + (-10.4 - 10.4i)T + 73iT^{2} \)
79 \( 1 - 13.1iT - 79T^{2} \)
83 \( 1 + (-5.76 + 5.76i)T - 83iT^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + (6.91 - 6.91i)T - 97iT^{2} \)
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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

−11.24442034876770159619300722573, −10.77016945758666160330913454522, −9.656306243621018949863133491750, −9.087857201070848731652990423645, −7.71550939187651938970260857011, −6.96027839264144736751780459639, −5.79231246348399298672219063086, −4.16085227855309817224790577894, −2.58212441661522093378193821347, −1.52716818507575925836757234439, 2.08190425512352371396647719257, 3.57923093320945749115951098759, 4.79147048293473680247101226384, 6.01530958426357082149786903960, 7.61948102909775845288940230940, 8.387077463683960989546206244850, 8.882556972003897326547524604009, 9.682995738073443043149087968191, 10.99747506974957067550931015058, 12.17024341124234980911405734895

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