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Properties

Label 2-585-9.7-c1-0-11
Degree $2$
Conductor $585$
Sign $-0.996 - 0.0788i$
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  + (−0.624 + 1.08i)2-s + (1.28 + 1.16i)3-s + (0.219 + 0.379i)4-s + (−0.5 − 0.866i)5-s + (−2.06 + 0.659i)6-s + (0.148 − 0.257i)7-s − 3.04·8-s + (0.286 + 2.98i)9-s + 1.24·10-s + (−1.85 + 3.20i)11-s + (−0.161 + 0.741i)12-s + (−0.5 − 0.866i)13-s + (0.185 + 0.321i)14-s + (0.367 − 1.69i)15-s + (1.46 − 2.53i)16-s − 3.47·17-s + ⋯
L(s)  = 1  + (−0.441 + 0.765i)2-s + (0.740 + 0.672i)3-s + (0.109 + 0.189i)4-s + (−0.223 − 0.387i)5-s + (−0.841 + 0.269i)6-s + (0.0562 − 0.0973i)7-s − 1.07·8-s + (0.0954 + 0.995i)9-s + 0.395·10-s + (−0.557 + 0.966i)11-s + (−0.0465 + 0.214i)12-s + (−0.138 − 0.240i)13-s + (0.0496 + 0.0860i)14-s + (0.0949 − 0.437i)15-s + (0.366 − 0.634i)16-s − 0.842·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.996 - 0.0788i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.996 - 0.0788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0452116 + 1.14547i\)
\(L(\frac12)\) \(\approx\) \(0.0452116 + 1.14547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.28 - 1.16i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.624 - 1.08i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.148 + 0.257i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.85 - 3.20i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.47T + 17T^{2} \)
19 \( 1 + 3.33T + 19T^{2} \)
23 \( 1 + (-4.29 - 7.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.94 - 5.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.44 - 5.96i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.40T + 37T^{2} \)
41 \( 1 + (5.14 + 8.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.74 + 4.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.77 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.01T + 53T^{2} \)
59 \( 1 + (-7.46 - 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.86 + 3.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.59 + 6.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 - 1.78T + 73T^{2} \)
79 \( 1 + (-0.213 + 0.369i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.05 + 3.55i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.14T + 89T^{2} \)
97 \( 1 + (-2.53 + 4.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.88606647346622736742441924217, −10.04746743776039204234077679610, −8.946291063991130849385577312838, −8.703016994803198198955720973983, −7.47108298644320895889831709010, −7.18108700084216137623029967302, −5.59431565406306221294827348038, −4.62202895798768233424096181391, −3.49323711000462517097806016816, −2.27271791927242523388127769691, 0.64757881202566143244974146085, 2.32688347094307509248280764551, 2.84842390999323525794732735798, 4.27914154614367167133600964545, 6.05960463600103930476798700295, 6.59671859255060614262368573242, 7.88151839542485070705914056389, 8.597347869791133614925057361648, 9.382365958726788818672877658486, 10.32934282242817483221835532447

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