Properties

Label 2-585-117.25-c1-0-24
Degree $2$
Conductor $585$
Sign $-0.109 + 0.993i$
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  + (−1.73 − i)2-s + (1.5 − 0.866i)3-s + (0.999 + 1.73i)4-s + (−0.866 + 0.5i)5-s − 3.46·6-s + (−1.73 − i)7-s + (1.5 − 2.59i)9-s + 1.99·10-s + (5.19 + 3i)11-s + (2.99 + 1.73i)12-s + (3.23 + 1.59i)13-s + (1.99 + 3.46i)14-s + (−0.866 + 1.5i)15-s + (1.99 − 3.46i)16-s − 17-s + (−5.19 + 3i)18-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.866 − 0.499i)3-s + (0.499 + 0.866i)4-s + (−0.387 + 0.223i)5-s − 1.41·6-s + (−0.654 − 0.377i)7-s + (0.5 − 0.866i)9-s + 0.632·10-s + (1.56 + 0.904i)11-s + (0.866 + 0.500i)12-s + (0.896 + 0.443i)13-s + (0.534 + 0.925i)14-s + (−0.223 + 0.387i)15-s + (0.499 − 0.866i)16-s − 0.242·17-s + (−1.22 + 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.109 + 0.993i$
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: \((2,\ 585,\ (\ :1/2),\ -0.109 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.664421 - 0.741941i\)
\(L(\frac12)\) \(\approx\) \(0.664421 - 0.741941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-3.23 - 1.59i)T \)
good2 \( 1 + (1.73 + i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.19 - 3i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.92 + 4i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (-1.73 + i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.66 + 5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-5.19 + 3i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10iT - 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-8.66 - 5i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.16485128636909388205433896882, −9.465300012981883236625980594521, −8.912190065310129832997154333236, −8.138730555944762309582064154136, −6.91133765759704696590064958901, −6.66431488351843470087323458027, −4.36964616414684454371753312904, −3.34617826599026920783086675271, −2.16296743276808025450618966523, −0.923451805834034125913335803134, 1.31908878548689850259178556920, 3.37431005155124049813486343894, 3.99733983962288186047700822684, 5.86129372743986573456806269248, 6.58438856348517467870276561944, 7.79301199580912705483573447502, 8.498938245570724661852554332540, 8.917788227400866796410289927992, 9.758130987141156448104424648202, 10.44850348129599342823979492884

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