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Properties

Label 2-285-15.2-c1-0-27
Degree $2$
Conductor $285$
Sign $-0.355 + 0.934i$
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  + (−0.0580 + 0.0580i)2-s + (0.605 − 1.62i)3-s + 1.99i·4-s + (−1.89 − 1.18i)5-s + (0.0590 + 0.129i)6-s + (−2.14 − 2.14i)7-s + (−0.231 − 0.231i)8-s + (−2.26 − 1.96i)9-s + (0.178 − 0.0409i)10-s − 4.16i·11-s + (3.23 + 1.20i)12-s + (2.98 − 2.98i)13-s + 0.248·14-s + (−3.07 + 2.35i)15-s − 3.95·16-s + (1.86 − 1.86i)17-s + ⋯
L(s)  = 1  + (−0.0410 + 0.0410i)2-s + (0.349 − 0.936i)3-s + 0.996i·4-s + (−0.847 − 0.531i)5-s + (0.0241 + 0.0527i)6-s + (−0.810 − 0.810i)7-s + (−0.0819 − 0.0819i)8-s + (−0.755 − 0.654i)9-s + (0.0565 − 0.0129i)10-s − 1.25i·11-s + (0.933 + 0.348i)12-s + (0.828 − 0.828i)13-s + 0.0665·14-s + (−0.794 + 0.607i)15-s − 0.989·16-s + (0.451 − 0.451i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.355 + 0.934i$
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: \((2,\ 285,\ (\ :1/2),\ -0.355 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.537693 - 0.780131i\)
\(L(\frac12)\) \(\approx\) \(0.537693 - 0.780131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.605 + 1.62i)T \)
5 \( 1 + (1.89 + 1.18i)T \)
19 \( 1 - iT \)
good2 \( 1 + (0.0580 - 0.0580i)T - 2iT^{2} \)
7 \( 1 + (2.14 + 2.14i)T + 7iT^{2} \)
11 \( 1 + 4.16iT - 11T^{2} \)
13 \( 1 + (-2.98 + 2.98i)T - 13iT^{2} \)
17 \( 1 + (-1.86 + 1.86i)T - 17iT^{2} \)
23 \( 1 + (-5.35 - 5.35i)T + 23iT^{2} \)
29 \( 1 + 5.48T + 29T^{2} \)
31 \( 1 + 0.743T + 31T^{2} \)
37 \( 1 + (3.93 + 3.93i)T + 37iT^{2} \)
41 \( 1 - 7.30iT - 41T^{2} \)
43 \( 1 + (-2.54 + 2.54i)T - 43iT^{2} \)
47 \( 1 + (-4.88 + 4.88i)T - 47iT^{2} \)
53 \( 1 + (-1.58 - 1.58i)T + 53iT^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 5.26T + 61T^{2} \)
67 \( 1 + (-1.58 - 1.58i)T + 67iT^{2} \)
71 \( 1 + 7.97iT - 71T^{2} \)
73 \( 1 + (-3.09 + 3.09i)T - 73iT^{2} \)
79 \( 1 - 4.68iT - 79T^{2} \)
83 \( 1 + (0.803 + 0.803i)T + 83iT^{2} \)
89 \( 1 - 7.94T + 89T^{2} \)
97 \( 1 + (-11.4 - 11.4i)T + 97iT^{2} \)
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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

−11.67432645701870383723775186676, −10.95528764298190895880353076022, −9.226626272362625703473685144394, −8.462866579425455725731100716852, −7.65926661644015912285488353023, −7.00857023993368491242035037982, −5.64892373577676048756711374744, −3.60630688011537614588106143878, −3.30326817771744340540805210614, −0.70338649911930054379001571758, 2.41116554642465742735739455322, 3.80234903631405778933432283792, 4.88285474657608464469674317590, 6.12396196333969127569506435254, 7.13199592192227650100442301703, 8.689455771483899011854940552410, 9.318406103941859922650504461143, 10.25900129916111001416731595835, 10.95520406711529417892187290090, 11.89114939269139608381880095672

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