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Properties

Label 2-585-117.25-c1-0-25
Degree $2$
Conductor $585$
Sign $0.174 + 0.984i$
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  + (−2.26 − 1.30i)2-s + (−0.538 + 1.64i)3-s + (2.41 + 4.18i)4-s + (0.866 − 0.5i)5-s + (3.37 − 3.02i)6-s + (3.14 + 1.81i)7-s − 7.40i·8-s + (−2.42 − 1.77i)9-s − 2.61·10-s + (−5.33 − 3.08i)11-s + (−8.19 + 1.72i)12-s + (0.233 − 3.59i)13-s + (−4.75 − 8.22i)14-s + (0.356 + 1.69i)15-s + (−4.84 + 8.39i)16-s + 2.59·17-s + ⋯
L(s)  = 1  + (−1.60 − 0.924i)2-s + (−0.310 + 0.950i)3-s + (1.20 + 2.09i)4-s + (0.387 − 0.223i)5-s + (1.37 − 1.23i)6-s + (1.18 + 0.687i)7-s − 2.61i·8-s + (−0.806 − 0.590i)9-s − 0.826·10-s + (−1.60 − 0.929i)11-s + (−2.36 + 0.497i)12-s + (0.0647 − 0.997i)13-s + (−1.26 − 2.19i)14-s + (0.0921 + 0.437i)15-s + (−1.21 + 2.09i)16-s + 0.630·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.174 + 0.984i$
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: \((2,\ 585,\ (\ :1/2),\ 0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431555 - 0.361645i\)
\(L(\frac12)\) \(\approx\) \(0.431555 - 0.361645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.538 - 1.64i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.233 + 3.59i)T \)
good2 \( 1 + (2.26 + 1.30i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-3.14 - 1.81i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.33 + 3.08i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.59T + 17T^{2} \)
19 \( 1 + 6.95iT - 19T^{2} \)
23 \( 1 + (-1.37 - 2.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.36 + 4.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.56 - 2.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.85iT - 37T^{2} \)
41 \( 1 + (-0.632 + 0.365i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.01 + 3.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0766 + 0.0442i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.07T + 53T^{2} \)
59 \( 1 + (-5.43 + 3.13i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.61 + 4.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.5 - 6.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.49iT - 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 + (2.01 - 3.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.4 - 7.21i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.71iT - 89T^{2} \)
97 \( 1 + (-12.5 - 7.22i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.59897058049184925189686772739, −9.732555599667646189000334121635, −8.811772842172796447567301462097, −8.352978202741212196843590895405, −7.51393144028645827166071445728, −5.68639903864830141761105740419, −5.00728814637491915489306156059, −3.21574266448070249921229337991, −2.39705365065392320062422505513, −0.59441925386786344925684116476, 1.33611952606962980211680651529, 2.16500806769417057157763564349, 4.91643834734636704945664891957, 5.78445348692747265762515487286, 6.79436082685769286085974063350, 7.67465012137076015648667088009, 7.82358046447415977553090914706, 8.865311274342500607394488954035, 10.13180114177845780520360351033, 10.53239357844891533588497885418

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