Properties

Label 2-585-65.38-c0-0-1
Degree $2$
Conductor $585$
Sign $-0.584 - 0.811i$
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  + (1.30 + 1.30i)2-s + 2.41i·4-s + (−0.923 + 0.382i)5-s + (−1.84 + 1.84i)8-s + (−1.70 − 0.707i)10-s − 1.84i·11-s + (0.707 + 0.707i)13-s − 2.41·16-s + (−0.923 − 2.23i)20-s + (2.41 − 2.41i)22-s + (0.707 − 0.707i)25-s + 1.84i·26-s + (−1.30 − 1.30i)32-s + (1.00 − 2.41i)40-s + 0.765i·41-s + ⋯
L(s)  = 1  + (1.30 + 1.30i)2-s + 2.41i·4-s + (−0.923 + 0.382i)5-s + (−1.84 + 1.84i)8-s + (−1.70 − 0.707i)10-s − 1.84i·11-s + (0.707 + 0.707i)13-s − 2.41·16-s + (−0.923 − 2.23i)20-s + (2.41 − 2.41i)22-s + (0.707 − 0.707i)25-s + 1.84i·26-s + (−1.30 − 1.30i)32-s + (1.00 − 2.41i)40-s + 0.765i·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(0.291953\)
Root analytic conductor: \(0.540326\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :0),\ -0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.496548709\)
\(L(\frac12)\) \(\approx\) \(1.496548709\)
\(L(1)\) 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 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + 1.84iT - T^{2} \)
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 \( 1 - 0.765iT - T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 0.765T + T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 0.765iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
89 \( 1 + 1.84T + T^{2} \)
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

−11.51698535593075887418137425205, −10.74190454206365760751838209808, −8.873055399219407593816508098266, −8.299254130005556637255294869504, −7.46539072737060528135118735333, −6.49936911497139519509939936504, −5.92832096469139046492801982925, −4.75563794454157060964909283617, −3.73894123633339661424945036085, −3.13614529021210741099343827663, 1.51692530949626808455739992371, 2.89638044487971571233836669821, 3.99095436727907082816390778127, 4.63467925979035227287303361122, 5.53254811583956369996033120175, 6.81751593913886566727291587568, 7.921363545392969964933473877695, 9.220004180320881461686229263664, 10.10957663507210970371288539767, 10.86695682967106755659582272211

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