Properties

Label 2-285-15.2-c1-0-13
Degree $2$
Conductor $285$
Sign $0.996 + 0.0857i$
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  + (−0.913 + 0.913i)2-s + (−0.951 − 1.44i)3-s + 0.331i·4-s + (2.10 − 0.765i)5-s + (2.19 + 0.452i)6-s + (0.194 + 0.194i)7-s + (−2.12 − 2.12i)8-s + (−1.18 + 2.75i)9-s + (−1.21 + 2.61i)10-s + 0.417i·11-s + (0.480 − 0.315i)12-s + (4.34 − 4.34i)13-s − 0.356·14-s + (−3.10 − 2.31i)15-s + 3.22·16-s + (1.75 − 1.75i)17-s + ⋯
L(s)  = 1  + (−0.645 + 0.645i)2-s + (−0.549 − 0.835i)3-s + 0.165i·4-s + (0.939 − 0.342i)5-s + (0.894 + 0.184i)6-s + (0.0736 + 0.0736i)7-s + (−0.752 − 0.752i)8-s + (−0.395 + 0.918i)9-s + (−0.385 + 0.827i)10-s + 0.125i·11-s + (0.138 − 0.0912i)12-s + (1.20 − 1.20i)13-s − 0.0951·14-s + (−0.802 − 0.596i)15-s + 0.806·16-s + (0.426 − 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.996 + 0.0857i$
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: \((2,\ 285,\ (\ :1/2),\ 0.996 + 0.0857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.906206 - 0.0389144i\)
\(L(\frac12)\) \(\approx\) \(0.906206 - 0.0389144i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 1.44i)T \)
5 \( 1 + (-2.10 + 0.765i)T \)
19 \( 1 + iT \)
good2 \( 1 + (0.913 - 0.913i)T - 2iT^{2} \)
7 \( 1 + (-0.194 - 0.194i)T + 7iT^{2} \)
11 \( 1 - 0.417iT - 11T^{2} \)
13 \( 1 + (-4.34 + 4.34i)T - 13iT^{2} \)
17 \( 1 + (-1.75 + 1.75i)T - 17iT^{2} \)
23 \( 1 + (-3.99 - 3.99i)T + 23iT^{2} \)
29 \( 1 - 7.49T + 29T^{2} \)
31 \( 1 + 1.04T + 31T^{2} \)
37 \( 1 + (3.18 + 3.18i)T + 37iT^{2} \)
41 \( 1 - 4.47iT - 41T^{2} \)
43 \( 1 + (-1.64 + 1.64i)T - 43iT^{2} \)
47 \( 1 + (6.22 - 6.22i)T - 47iT^{2} \)
53 \( 1 + (7.09 + 7.09i)T + 53iT^{2} \)
59 \( 1 + 4.72T + 59T^{2} \)
61 \( 1 - 0.494T + 61T^{2} \)
67 \( 1 + (0.00945 + 0.00945i)T + 67iT^{2} \)
71 \( 1 - 6.32iT - 71T^{2} \)
73 \( 1 + (10.3 - 10.3i)T - 73iT^{2} \)
79 \( 1 - 3.38iT - 79T^{2} \)
83 \( 1 + (4.89 + 4.89i)T + 83iT^{2} \)
89 \( 1 - 2.98T + 89T^{2} \)
97 \( 1 + (-6.77 - 6.77i)T + 97iT^{2} \)
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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

−11.92638184660247836385477566355, −10.82017703799633943180363622086, −9.770387809039420754477863094877, −8.677361340319229122469203557686, −7.986271747543163413565663551245, −6.91210735418182478443021845094, −6.06414232955622068869121585694, −5.18401639241933116027099593235, −3.02177555855043840272019544370, −1.09880033013297305551532937587, 1.44409833026144790056387555317, 3.12172789118915386944765202362, 4.70150119076343700661427724616, 5.93194198800764073914429232881, 6.53949535230709194745598402991, 8.634286445920441223356009505382, 9.201312359265875253330060654503, 10.17518168573806335629976479751, 10.70436839133566232202400034302, 11.40176562109140781364147263901

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