Properties

Label 2-285-15.2-c1-0-25
Degree $2$
Conductor $285$
Sign $0.828 + 0.559i$
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  + (1.43 − 1.43i)2-s + (0.368 + 1.69i)3-s − 2.13i·4-s + (2.18 − 0.464i)5-s + (2.96 + 1.90i)6-s + (−1.23 − 1.23i)7-s + (−0.197 − 0.197i)8-s + (−2.72 + 1.24i)9-s + (2.47 − 3.81i)10-s − 0.351i·11-s + (3.61 − 0.786i)12-s + (0.181 − 0.181i)13-s − 3.54·14-s + (1.59 + 3.53i)15-s + 3.70·16-s + (−1.52 + 1.52i)17-s + ⋯
L(s)  = 1  + (1.01 − 1.01i)2-s + (0.212 + 0.977i)3-s − 1.06i·4-s + (0.978 − 0.207i)5-s + (1.20 + 0.777i)6-s + (−0.465 − 0.465i)7-s + (−0.0698 − 0.0698i)8-s + (−0.909 + 0.415i)9-s + (0.783 − 1.20i)10-s − 0.105i·11-s + (1.04 − 0.227i)12-s + (0.0502 − 0.0502i)13-s − 0.947·14-s + (0.410 + 0.911i)15-s + 0.926·16-s + (−0.369 + 0.369i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.828 + 0.559i$
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: \((2,\ 285,\ (\ :1/2),\ 0.828 + 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.30539 - 0.705471i\)
\(L(\frac12)\) \(\approx\) \(2.30539 - 0.705471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.368 - 1.69i)T \)
5 \( 1 + (-2.18 + 0.464i)T \)
19 \( 1 - iT \)
good2 \( 1 + (-1.43 + 1.43i)T - 2iT^{2} \)
7 \( 1 + (1.23 + 1.23i)T + 7iT^{2} \)
11 \( 1 + 0.351iT - 11T^{2} \)
13 \( 1 + (-0.181 + 0.181i)T - 13iT^{2} \)
17 \( 1 + (1.52 - 1.52i)T - 17iT^{2} \)
23 \( 1 + (3.86 + 3.86i)T + 23iT^{2} \)
29 \( 1 + 7.70T + 29T^{2} \)
31 \( 1 + 4.22T + 31T^{2} \)
37 \( 1 + (-1.89 - 1.89i)T + 37iT^{2} \)
41 \( 1 - 6.12iT - 41T^{2} \)
43 \( 1 + (0.226 - 0.226i)T - 43iT^{2} \)
47 \( 1 + (8.19 - 8.19i)T - 47iT^{2} \)
53 \( 1 + (2.43 + 2.43i)T + 53iT^{2} \)
59 \( 1 - 5.37T + 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + (5.18 + 5.18i)T + 67iT^{2} \)
71 \( 1 - 14.9iT - 71T^{2} \)
73 \( 1 + (-6.94 + 6.94i)T - 73iT^{2} \)
79 \( 1 + 3.78iT - 79T^{2} \)
83 \( 1 + (7.19 + 7.19i)T + 83iT^{2} \)
89 \( 1 - 17.9T + 89T^{2} \)
97 \( 1 + (-9.20 - 9.20i)T + 97iT^{2} \)
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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

−11.65867867801782881357151823897, −10.78154369119791539973530076144, −10.11393765311906106355604429242, −9.365799386179202088798948906761, −8.128842225523824816011077267788, −6.27466590901485105337123163915, −5.30975216504794551956293342513, −4.29421448924202913985597485485, −3.33439355985729816141816610577, −2.08254688439338175909587170501, 2.07017587861855963294428324092, 3.53560133343834697333654679069, 5.32039328091918290724406873181, 5.95568820272005128784223738887, 6.80816765089243177785723163974, 7.56485057335514457559938770541, 8.886978142903232932526820176537, 9.819664209171393899038345796600, 11.28463942506315814817678546150, 12.44888339258850425208930132793

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