Properties

Label 2-585-13.9-c1-0-5
Degree $2$
Conductor $585$
Sign $-0.798 - 0.601i$
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  + (−0.905 + 1.56i)2-s + (−0.638 − 1.10i)4-s + 5-s + (2.21 + 3.83i)7-s − 1.30·8-s + (−0.905 + 1.56i)10-s + (0.807 − 1.39i)11-s + (3.35 + 1.32i)13-s − 8.01·14-s + (2.46 − 4.26i)16-s + (0.627 + 1.08i)17-s + (0.361 + 0.625i)19-s + (−0.638 − 1.10i)20-s + (1.46 + 2.53i)22-s + (−0.654 + 1.13i)23-s + ⋯
L(s)  = 1  + (−0.640 + 1.10i)2-s + (−0.319 − 0.553i)4-s + 0.447·5-s + (0.836 + 1.44i)7-s − 0.462·8-s + (−0.286 + 0.495i)10-s + (0.243 − 0.421i)11-s + (0.929 + 0.368i)13-s − 2.14·14-s + (0.615 − 1.06i)16-s + (0.152 + 0.263i)17-s + (0.0829 + 0.143i)19-s + (−0.142 − 0.247i)20-s + (0.311 + 0.539i)22-s + (−0.136 + 0.236i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.798 - 0.601i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.798 - 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383622 + 1.14738i\)
\(L(\frac12)\) \(\approx\) \(0.383622 + 1.14738i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (-3.35 - 1.32i)T \)
good2 \( 1 + (0.905 - 1.56i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-2.21 - 3.83i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.807 + 1.39i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.627 - 1.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.361 - 0.625i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.654 - 1.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.580 - 1.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.64T + 31T^{2} \)
37 \( 1 + (2.46 - 4.26i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.28 - 7.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.57 + 6.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + (2.14 + 3.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.82 + 4.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.109 - 0.189i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.92 + 5.05i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.91T + 73T^{2} \)
79 \( 1 + 14.1T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 + (-0.422 + 0.732i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.77 + 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.07517625990856119857034570899, −9.794401791252721474377648985490, −8.856519430606442018023125098976, −8.562070936376074590278003257696, −7.65448638653880879911071030035, −6.40605176180359427248092905982, −5.88489727588578262204875568977, −5.00138174805353059991115568182, −3.24520787399199107961889446990, −1.74044873838575182690598360895, 0.917692887703527879561682108541, 1.91030158703125581405398973645, 3.39197185535506455895658665966, 4.39746882253493348749387109638, 5.72600508951836652369981684163, 6.91816423809028229626782114682, 7.895241719696889394994112889039, 8.845982187637741345387487460804, 9.712314336136825906469484580630, 10.59191709117856714636741325650

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