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Properties

Label 2-585-65.12-c0-0-0
Degree $2$
Conductor $585$
Sign $-0.811 - 0.584i$
Analytic cond. $0.291953$
Root an. cond. $0.540326$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.541 + 0.541i)2-s + 0.414i·4-s + (−0.382 + 0.923i)5-s + (−0.765 − 0.765i)8-s + (−0.292 − 0.707i)10-s + 0.765i·11-s + (−0.707 + 0.707i)13-s + 0.414·16-s + (−0.382 − 0.158i)20-s + (−0.414 − 0.414i)22-s + (−0.707 − 0.707i)25-s − 0.765i·26-s + (0.541 − 0.541i)32-s + (1 − 0.414i)40-s + 1.84i·41-s + ⋯
L(s)  = 1  + (−0.541 + 0.541i)2-s + 0.414i·4-s + (−0.382 + 0.923i)5-s + (−0.765 − 0.765i)8-s + (−0.292 − 0.707i)10-s + 0.765i·11-s + (−0.707 + 0.707i)13-s + 0.414·16-s + (−0.382 − 0.158i)20-s + (−0.414 − 0.414i)22-s + (−0.707 − 0.707i)25-s − 0.765i·26-s + (0.541 − 0.541i)32-s + (1 − 0.414i)40-s + 1.84i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.811 - 0.584i$
Analytic conductor: \(0.291953\)
Root analytic conductor: \(0.540326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (532, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :0),\ -0.811 - 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5434603687\)
\(L(\frac12)\) \(\approx\) \(0.5434603687\)
\(L(1)\) 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 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - 0.765iT - T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.84T + T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.84iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
89 \( 1 + 0.765T + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−11.34553975704963622605779311378, −10.11881645754644177956993619880, −9.545179143707910087542589284762, −8.467868407815004936532586641882, −7.59264461117258332536905574925, −6.99829100007556552456065230010, −6.26655755857492201074577143831, −4.66997268944348490895000365137, −3.59008916874264094497740478474, −2.41460360310530724210722720962, 0.76263624706000351682124792638, 2.32369453557618542888148133738, 3.75146405964570847438051736670, 5.16152028384679902468942590721, 5.69818673999793289560039417269, 7.13968050604446613271019429800, 8.318435452824471949653106141041, 8.799854915718864510827157696324, 9.762751858319427314569905978455, 10.47491852799362796014399304232

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