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Properties

Label 2-585-65.29-c1-0-24
Degree $2$
Conductor $585$
Sign $0.778 + 0.627i$
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  + (1.57 − 0.907i)2-s + (0.645 − 1.11i)4-s + (1.19 − 1.88i)5-s + (4.06 + 2.34i)7-s + 1.28i·8-s + (0.167 − 4.05i)10-s + (0.270 + 0.468i)11-s + (−1.34 + 3.34i)13-s + 8.50·14-s + (2.45 + 4.25i)16-s + (−5.92 − 3.42i)17-s + (2.78 − 4.81i)19-s + (−1.34 − 2.55i)20-s + (0.849 + 0.490i)22-s + (0.0291 − 0.0168i)23-s + ⋯
L(s)  = 1  + (1.11 − 0.641i)2-s + (0.322 − 0.559i)4-s + (0.535 − 0.844i)5-s + (1.53 + 0.886i)7-s + 0.454i·8-s + (0.0528 − 1.28i)10-s + (0.0815 + 0.141i)11-s + (−0.372 + 0.928i)13-s + 2.27·14-s + (0.614 + 1.06i)16-s + (−1.43 − 0.829i)17-s + (0.638 − 1.10i)19-s + (−0.299 − 0.572i)20-s + (0.181 + 0.104i)22-s + (0.00607 − 0.00350i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.778 + 0.627i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.778 + 0.627i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.87511 - 1.01393i\)
\(L(\frac12)\) \(\approx\) \(2.87511 - 1.01393i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.19 + 1.88i)T \)
13 \( 1 + (1.34 - 3.34i)T \)
good2 \( 1 + (-1.57 + 0.907i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-4.06 - 2.34i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.270 - 0.468i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (5.92 + 3.42i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.78 + 4.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0291 + 0.0168i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.40 + 5.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.352T + 31T^{2} \)
37 \( 1 + (6.98 - 4.03i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.59 - 6.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.08 + 3.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.99iT - 47T^{2} \)
53 \( 1 + 8.50iT - 53T^{2} \)
59 \( 1 + (-6.64 + 11.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.60 + 2.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.70 - 5.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.33 - 7.50i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.05iT - 73T^{2} \)
79 \( 1 + 2.29T + 79T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 + (-3.98 - 6.90i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.265 - 0.153i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.27769793080028555298376667706, −9.683578548576363367990657159440, −8.850874125668933956799477940336, −8.216424947174379792670286013715, −6.79381322527316207190513584627, −5.41510016162149715954554060660, −4.92126504149400426606998217583, −4.29076636415629979212168890394, −2.47709331066294558179482032302, −1.81937038599821088110693061265, 1.70200234238720715839493719448, 3.37904838878767440593684489929, 4.34343883350093135777049884148, 5.31247458934525866567591804958, 6.05262669392238536829283621075, 7.19888187963228330754823174477, 7.64182799077229314094091338677, 8.914996610298361536252678802540, 10.42238687382002489316478287116, 10.57992316322046992049071060189

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