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Properties

Label 2-585-9.4-c1-0-9
Degree $2$
Conductor $585$
Sign $-0.810 - 0.585i$
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  + (0.984 + 1.70i)2-s + (−1.60 + 0.651i)3-s + (−0.938 + 1.62i)4-s + (0.5 − 0.866i)5-s + (−2.69 − 2.09i)6-s + (1.51 + 2.62i)7-s + 0.240·8-s + (2.15 − 2.09i)9-s + 1.96·10-s + (2.15 + 3.72i)11-s + (0.447 − 3.22i)12-s + (0.5 − 0.866i)13-s + (−2.98 + 5.16i)14-s + (−0.238 + 1.71i)15-s + (2.11 + 3.66i)16-s − 0.303·17-s + ⋯
L(s)  = 1  + (0.696 + 1.20i)2-s + (−0.926 + 0.376i)3-s + (−0.469 + 0.813i)4-s + (0.223 − 0.387i)5-s + (−1.09 − 0.855i)6-s + (0.572 + 0.991i)7-s + 0.0850·8-s + (0.717 − 0.696i)9-s + 0.622·10-s + (0.648 + 1.12i)11-s + (0.129 − 0.929i)12-s + (0.138 − 0.240i)13-s + (−0.796 + 1.38i)14-s + (−0.0615 + 0.442i)15-s + (0.528 + 0.915i)16-s − 0.0735·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.810 - 0.585i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.810 - 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.547241 + 1.69294i\)
\(L(\frac12)\) \(\approx\) \(0.547241 + 1.69294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 - 0.651i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.984 - 1.70i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.51 - 2.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.15 - 3.72i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.303T + 17T^{2} \)
19 \( 1 + 6.04T + 19T^{2} \)
23 \( 1 + (1.47 - 2.55i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.44 + 7.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0319 - 0.0554i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.1T + 37T^{2} \)
41 \( 1 + (4.09 - 7.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.45 + 2.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.44 - 5.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + (4.57 - 7.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.657 + 1.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.91 + 13.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.85T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + (5.17 + 8.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.53 - 6.12i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.26T + 89T^{2} \)
97 \( 1 + (2.91 + 5.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.22933417582480293623541355950, −10.10890284123100369826929464063, −9.247466537216126620325839816376, −8.196231120476926862929209528501, −7.21005777628561604735528299987, −6.12495035042488842243630772819, −5.77261253003458787274365435880, −4.65504163283253373242330401085, −4.21211629323220733795683377391, −1.85450338786010052387081451929, 0.988359491574311493623613773316, 2.15926971738357951998811209019, 3.73134254836445658753717810415, 4.42432110873683508412029387845, 5.57961187225730854382086536986, 6.58495635636726968403472151201, 7.45742434656417465252846311196, 8.674588460365161963995009767233, 10.14546085032234942624428014962, 10.74625892046088372644149536473

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