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Properties

Label 2-585-117.103-c1-0-32
Degree $2$
Conductor $585$
Sign $0.999 - 0.00176i$
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  + (0.569 − 0.328i)2-s + (1.49 − 0.880i)3-s + (−0.783 + 1.35i)4-s + (0.866 + 0.5i)5-s + (0.560 − 0.991i)6-s + (0.282 − 0.163i)7-s + 2.34i·8-s + (1.44 − 2.62i)9-s + 0.657·10-s + (−1.86 + 1.07i)11-s + (0.0260 + 2.71i)12-s + (3.53 + 0.687i)13-s + (0.107 − 0.185i)14-s + (1.73 − 0.0166i)15-s + (−0.796 − 1.37i)16-s + 7.02·17-s + ⋯
L(s)  = 1  + (0.402 − 0.232i)2-s + (0.861 − 0.508i)3-s + (−0.391 + 0.678i)4-s + (0.387 + 0.223i)5-s + (0.228 − 0.404i)6-s + (0.106 − 0.0616i)7-s + 0.829i·8-s + (0.483 − 0.875i)9-s + 0.207·10-s + (−0.563 + 0.325i)11-s + (0.00751 + 0.783i)12-s + (0.981 + 0.190i)13-s + (0.0286 − 0.0496i)14-s + (0.447 − 0.00428i)15-s + (−0.199 − 0.344i)16-s + 1.70·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.999 - 0.00176i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.999 - 0.00176i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39896 + 0.00211602i\)
\(L(\frac12)\) \(\approx\) \(2.39896 + 0.00211602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.49 + 0.880i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-3.53 - 0.687i)T \)
good2 \( 1 + (-0.569 + 0.328i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.282 + 0.163i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.86 - 1.07i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 7.02T + 17T^{2} \)
19 \( 1 - 1.98iT - 19T^{2} \)
23 \( 1 + (-1.07 + 1.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.80 - 3.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.45 + 3.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.86iT - 37T^{2} \)
41 \( 1 + (3.36 + 1.94i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.83 + 4.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0831 + 0.0480i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.33T + 53T^{2} \)
59 \( 1 + (-8.49 - 4.90i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.60 + 13.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.62 + 2.67i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 + (5.54 + 9.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (13.8 - 7.99i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.29iT - 89T^{2} \)
97 \( 1 + (-1.55 + 0.897i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.68261107706543503979531188497, −9.728467612572264753830651109391, −8.815034860595220650044883504044, −8.026229651181710570429630959914, −7.39394319419130973771743905767, −6.16958980304353313005069649325, −4.99534576138112207570574337646, −3.67782494586653185141630084333, −3.03101874104356702372535317024, −1.68469348224833892843609403922, 1.40109840462215866497202824093, 3.08248918354171239288192221421, 4.06943897438424855389077537808, 5.26801517126943576596857698248, 5.73646107274905537978596900115, 7.15245206652413533371727360844, 8.254707705080772243680027022753, 8.965511400515594074754309525187, 9.878482774518329494033262861881, 10.37695251878665238916317102243

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