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Properties

Label 2-585-117.103-c1-0-1
Degree $2$
Conductor $585$
Sign $0.191 + 0.981i$
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  + (−2.20 + 1.27i)2-s + (0.350 + 1.69i)3-s + (2.23 − 3.87i)4-s + (−0.866 − 0.5i)5-s + (−2.92 − 3.29i)6-s + (−0.586 + 0.338i)7-s + 6.28i·8-s + (−2.75 + 1.18i)9-s + 2.54·10-s + (−3.28 + 1.89i)11-s + (7.35 + 2.43i)12-s + (2.20 + 2.85i)13-s + (0.861 − 1.49i)14-s + (0.544 − 1.64i)15-s + (−3.52 − 6.10i)16-s + 0.607·17-s + ⋯
L(s)  = 1  + (−1.55 + 0.899i)2-s + (0.202 + 0.979i)3-s + (1.11 − 1.93i)4-s + (−0.387 − 0.223i)5-s + (−1.19 − 1.34i)6-s + (−0.221 + 0.127i)7-s + 2.22i·8-s + (−0.918 + 0.396i)9-s + 0.804·10-s + (−0.991 + 0.572i)11-s + (2.12 + 0.703i)12-s + (0.611 + 0.791i)13-s + (0.230 − 0.398i)14-s + (0.140 − 0.424i)15-s + (−0.881 − 1.52i)16-s + 0.147·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.191 + 0.981i$
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: \((2,\ 585,\ (\ :1/2),\ 0.191 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0285621 - 0.0235382i\)
\(L(\frac12)\) \(\approx\) \(0.0285621 - 0.0235382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.350 - 1.69i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-2.20 - 2.85i)T \)
good2 \( 1 + (2.20 - 1.27i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.586 - 0.338i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.28 - 1.89i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.607T + 17T^{2} \)
19 \( 1 + 3.20iT - 19T^{2} \)
23 \( 1 + (2.41 - 4.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.96 + 6.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.27 + 3.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.77iT - 37T^{2} \)
41 \( 1 + (-3.24 - 1.87i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.70 + 9.88i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.36 + 4.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.76T + 53T^{2} \)
59 \( 1 + (2.31 + 1.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.86 + 3.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.92 - 1.68i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.87iT - 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + (2.46 + 4.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.1 - 5.88i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 + (4.48 - 2.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.98764909000705797022540569885, −10.15797082423087414009951893192, −9.411925147623236893913615072716, −8.909281150022433635947162155369, −7.944942523893876710076904245416, −7.33494686042638689286451687562, −6.06874477808841798588039996682, −5.21613637287819380043957595737, −3.91133129687035421196753066655, −2.16360674861246409573391942101, 0.03392149698190848900354842684, 1.41238877975415034102922596301, 2.77489896749691386638138772757, 3.46843360219290541966959168449, 5.70540842597364380712043343625, 6.88385427468876372879174992603, 7.79966744457587514837827872540, 8.228143454556015573121589154311, 8.996947843622683834511671832650, 10.16231311098424784416969444174

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