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Properties

Label 2-285-15.2-c1-0-20
Degree $2$
Conductor $285$
Sign $0.478 + 0.877i$
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  + (0.885 − 0.885i)2-s + (−1.72 + 0.193i)3-s + 0.432i·4-s + (0.370 − 2.20i)5-s + (−1.35 + 1.69i)6-s + (0.957 + 0.957i)7-s + (2.15 + 2.15i)8-s + (2.92 − 0.666i)9-s + (−1.62 − 2.28i)10-s − 5.76i·11-s + (−0.0837 − 0.743i)12-s + (4.19 − 4.19i)13-s + 1.69·14-s + (−0.209 + 3.86i)15-s + 2.94·16-s + (−2.99 + 2.99i)17-s + ⋯
L(s)  = 1  + (0.626 − 0.626i)2-s + (−0.993 + 0.111i)3-s + 0.216i·4-s + (0.165 − 0.986i)5-s + (−0.552 + 0.692i)6-s + (0.362 + 0.362i)7-s + (0.761 + 0.761i)8-s + (0.974 − 0.222i)9-s + (−0.513 − 0.721i)10-s − 1.73i·11-s + (−0.0241 − 0.214i)12-s + (1.16 − 1.16i)13-s + 0.453·14-s + (−0.0541 + 0.998i)15-s + 0.737·16-s + (−0.726 + 0.726i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.478 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.478 + 0.877i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.478 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25909 - 0.747388i\)
\(L(\frac12)\) \(\approx\) \(1.25909 - 0.747388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.193i)T \)
5 \( 1 + (-0.370 + 2.20i)T \)
19 \( 1 - iT \)
good2 \( 1 + (-0.885 + 0.885i)T - 2iT^{2} \)
7 \( 1 + (-0.957 - 0.957i)T + 7iT^{2} \)
11 \( 1 + 5.76iT - 11T^{2} \)
13 \( 1 + (-4.19 + 4.19i)T - 13iT^{2} \)
17 \( 1 + (2.99 - 2.99i)T - 17iT^{2} \)
23 \( 1 + (-1.13 - 1.13i)T + 23iT^{2} \)
29 \( 1 - 5.41T + 29T^{2} \)
31 \( 1 + 0.142T + 31T^{2} \)
37 \( 1 + (-6.07 - 6.07i)T + 37iT^{2} \)
41 \( 1 + 3.96iT - 41T^{2} \)
43 \( 1 + (7.73 - 7.73i)T - 43iT^{2} \)
47 \( 1 + (4.30 - 4.30i)T - 47iT^{2} \)
53 \( 1 + (1.94 + 1.94i)T + 53iT^{2} \)
59 \( 1 + 6.82T + 59T^{2} \)
61 \( 1 - 3.11T + 61T^{2} \)
67 \( 1 + (-1.30 - 1.30i)T + 67iT^{2} \)
71 \( 1 - 2.68iT - 71T^{2} \)
73 \( 1 + (8.18 - 8.18i)T - 73iT^{2} \)
79 \( 1 - 11.2iT - 79T^{2} \)
83 \( 1 + (4.47 + 4.47i)T + 83iT^{2} \)
89 \( 1 + 5.91T + 89T^{2} \)
97 \( 1 + (2.71 + 2.71i)T + 97iT^{2} \)
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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

−11.47111112630328574617024183397, −11.24270824746206619723783022498, −10.18085392306748002978223022261, −8.532685779800224277717879161666, −8.192862117662667421269470254433, −6.19060359284915200851581934366, −5.48438680015101145937202615842, −4.47223293000696773340907219054, −3.29968924401564536839879214982, −1.24911458605027832529172541516, 1.78559492807430950683927454388, 4.20589653585576981723986504639, 4.86003565820496750191662437347, 6.24574835109797304930654396148, 6.80863407132785877340726791927, 7.44149548462323485788099151914, 9.432148721808474828994704990492, 10.33189960335819489742497885823, 11.04871398384765648818209903636, 11.89310801178731100070016587369

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