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Properties

Label 2-585-117.103-c1-0-12
Degree $2$
Conductor $585$
Sign $-0.842 - 0.538i$
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  + (0.774 − 0.447i)2-s + (−0.513 + 1.65i)3-s + (−0.600 + 1.03i)4-s + (0.866 + 0.5i)5-s + (0.342 + 1.51i)6-s + (−1.76 + 1.02i)7-s + 2.86i·8-s + (−2.47 − 1.69i)9-s + 0.894·10-s + (−0.659 + 0.380i)11-s + (−1.41 − 1.52i)12-s + (2.22 − 2.83i)13-s + (−0.913 + 1.58i)14-s + (−1.27 + 1.17i)15-s + (0.0784 + 0.135i)16-s − 2.67·17-s + ⋯
L(s)  = 1  + (0.547 − 0.316i)2-s + (−0.296 + 0.955i)3-s + (−0.300 + 0.519i)4-s + (0.387 + 0.223i)5-s + (0.139 + 0.616i)6-s + (−0.668 + 0.386i)7-s + 1.01i·8-s + (−0.824 − 0.566i)9-s + 0.282·10-s + (−0.198 + 0.114i)11-s + (−0.407 − 0.440i)12-s + (0.617 − 0.786i)13-s + (−0.244 + 0.422i)14-s + (−0.328 + 0.303i)15-s + (0.0196 + 0.0339i)16-s − 0.649·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.842 - 0.538i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.317007 + 1.08481i\)
\(L(\frac12)\) \(\approx\) \(0.317007 + 1.08481i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.513 - 1.65i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-2.22 + 2.83i)T \)
good2 \( 1 + (-0.774 + 0.447i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (1.76 - 1.02i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.659 - 0.380i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 2.67T + 17T^{2} \)
19 \( 1 - 7.38iT - 19T^{2} \)
23 \( 1 + (1.72 - 2.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.28 + 7.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.28 + 4.20i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.24iT - 37T^{2} \)
41 \( 1 + (-5.25 - 3.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.42 - 9.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.719 - 0.415i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + (-8.57 - 4.95i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.11 - 1.93i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.874 - 0.504i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.2iT - 71T^{2} \)
73 \( 1 - 8.99iT - 73T^{2} \)
79 \( 1 + (7.11 + 12.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.52 + 4.34i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.84iT - 89T^{2} \)
97 \( 1 + (0.0788 - 0.0455i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.18066089748280814382764060008, −10.15276785823919081042501907317, −9.513540258175489597205471443412, −8.577420403012260352318993015613, −7.65034000342232078996078869025, −5.87216230411810691133438601077, −5.73129626084259629821203632577, −4.23699168296292480238055527071, −3.56625014960150454277496758936, −2.50254754617512535877533249328, 0.53797945835299429072061995733, 2.08213550634002877026332299074, 3.74779599827455488760940751650, 5.00112327687537456945565855374, 5.78274270947612113503959819991, 6.81059174270818740434159463468, 7.05826978092624943395413086612, 8.816040991691338914847785352779, 9.194254625346483042916920968652, 10.58763229492687339669661243010

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