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Properties

Label 2-585-117.25-c1-0-1
Degree $2$
Conductor $585$
Sign $0.0928 - 0.995i$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
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  + (−1.85 − 1.07i)2-s + (−1.70 + 0.297i)3-s + (1.29 + 2.23i)4-s + (0.866 − 0.5i)5-s + (3.48 + 1.27i)6-s + (−4.18 − 2.41i)7-s − 1.24i·8-s + (2.82 − 1.01i)9-s − 2.14·10-s + (−0.777 − 0.448i)11-s + (−2.87 − 3.43i)12-s + (3.48 − 0.925i)13-s + (5.17 + 8.96i)14-s + (−1.32 + 1.11i)15-s + (1.24 − 2.15i)16-s − 5.40·17-s + ⋯
L(s)  = 1  + (−1.31 − 0.756i)2-s + (−0.985 + 0.171i)3-s + (0.645 + 1.11i)4-s + (0.387 − 0.223i)5-s + (1.42 + 0.520i)6-s + (−1.58 − 0.913i)7-s − 0.441i·8-s + (0.940 − 0.338i)9-s − 0.677·10-s + (−0.234 − 0.135i)11-s + (−0.828 − 0.991i)12-s + (0.966 − 0.256i)13-s + (1.38 + 2.39i)14-s + (−0.343 + 0.286i)15-s + (0.311 − 0.539i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0928 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0928 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.0928 - 0.995i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.0928 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0265727 + 0.0242104i\)
\(L(\frac12)\) \(\approx\) \(0.0265727 + 0.0242104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.297i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-3.48 + 0.925i)T \)
good2 \( 1 + (1.85 + 1.07i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (4.18 + 2.41i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.777 + 0.448i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.40T + 17T^{2} \)
19 \( 1 + 6.78iT - 19T^{2} \)
23 \( 1 + (-0.544 - 0.943i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.28 - 2.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.42 - 2.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.86iT - 37T^{2} \)
41 \( 1 + (3.50 - 2.02i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.08 + 3.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.64 - 3.83i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.85T + 53T^{2} \)
59 \( 1 + (13.0 - 7.55i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.04 - 5.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.26 + 3.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + (1.01 - 1.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.21 + 0.700i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.44iT - 89T^{2} \)
97 \( 1 + (15.9 + 9.23i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.80694288283752669291657046659, −10.14975795281705901043925473160, −9.314563120288592986015163080734, −8.791354119535471238001354969592, −7.22299874797572061112865638426, −6.65661549463330571175126584791, −5.53914960058629902116508039010, −4.12568968112760080445862103926, −2.85130636392665584449219174714, −1.09263683166803443507752575617, 0.04140698656280195061128951307, 1.94673837807131833663351946752, 3.79042148859993916539568311767, 5.63862816092594537361575940933, 6.24359778760225735708640026750, 6.69413570241673537446549031646, 7.77752757064352785036866533345, 8.946196469867433210449466577049, 9.472764310882774908419145019007, 10.32203018210955691751481311559

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