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Properties

Label 2-585-13.4-c1-0-5
Degree $2$
Conductor $585$
Sign $0.939 - 0.343i$
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  + (−2.10 − 1.21i)2-s + (1.95 + 3.38i)4-s + i·5-s + (1.12 − 0.650i)7-s − 4.64i·8-s + (1.21 − 2.10i)10-s + (2.75 + 1.58i)11-s + (−3.40 + 1.19i)13-s − 3.16·14-s + (−1.73 + 3.00i)16-s + (−0.325 − 0.564i)17-s + (2.17 − 1.25i)19-s + (−3.38 + 1.95i)20-s + (−3.86 − 6.68i)22-s + (−0.889 + 1.54i)23-s + ⋯
L(s)  = 1  + (−1.48 − 0.859i)2-s + (0.977 + 1.69i)4-s + 0.447i·5-s + (0.425 − 0.245i)7-s − 1.64i·8-s + (0.384 − 0.665i)10-s + (0.829 + 0.478i)11-s + (−0.943 + 0.331i)13-s − 0.845·14-s + (−0.433 + 0.750i)16-s + (−0.0789 − 0.136i)17-s + (0.499 − 0.288i)19-s + (−0.757 + 0.437i)20-s + (−0.823 − 1.42i)22-s + (−0.185 + 0.321i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.939 - 0.343i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.939 - 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.645950 + 0.114313i\)
\(L(\frac12)\) \(\approx\) \(0.645950 + 0.114313i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (3.40 - 1.19i)T \)
good2 \( 1 + (2.10 + 1.21i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-1.12 + 0.650i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.75 - 1.58i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.325 + 0.564i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.17 + 1.25i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.889 - 1.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.18 - 7.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.50iT - 31T^{2} \)
37 \( 1 + (-1.21 - 0.698i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.0 - 5.81i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.63 - 2.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 - 6.35T + 53T^{2} \)
59 \( 1 + (-5.18 + 2.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.49 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.3 - 6.57i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.949 + 0.548i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.2iT - 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 - 14.2iT - 83T^{2} \)
89 \( 1 + (-8.31 - 4.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.88 + 3.97i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.71209325992222187191028989093, −9.740256477917361426275054163973, −9.310333825754731363719811996213, −8.306812082189701175105473611980, −7.32836814252305386897533635983, −6.83789112786532683842464945669, −5.07209232214028894521058475030, −3.67395507013741730203385062298, −2.46235265371588143854540600273, −1.31909398621153233467810022130, 0.67142412409721672321213575336, 2.15821005581519526573552304846, 4.16739347809384067770418421889, 5.58822969740905675287398209949, 6.22078651867461874948113452965, 7.53096519073713157718248000279, 7.889152972538004479943552334694, 8.987610759234063385065276451610, 9.425852391452293058334803782100, 10.30437633092129544960622908153

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