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Properties

Label 2-585-9.4-c1-0-16
Degree $2$
Conductor $585$
Sign $-0.152 - 0.988i$
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  + (0.562 + 0.973i)2-s + (−0.0186 + 1.73i)3-s + (0.368 − 0.637i)4-s + (0.5 − 0.866i)5-s + (−1.69 + 0.955i)6-s + (1.91 + 3.31i)7-s + 3.07·8-s + (−2.99 − 0.0646i)9-s + 1.12·10-s + (−0.975 − 1.68i)11-s + (1.09 + 0.649i)12-s + (−0.5 + 0.866i)13-s + (−2.15 + 3.72i)14-s + (1.49 + 0.882i)15-s + (0.992 + 1.71i)16-s + 6.58·17-s + ⋯
L(s)  = 1  + (0.397 + 0.688i)2-s + (−0.0107 + 0.999i)3-s + (0.184 − 0.318i)4-s + (0.223 − 0.387i)5-s + (−0.692 + 0.389i)6-s + (0.723 + 1.25i)7-s + 1.08·8-s + (−0.999 − 0.0215i)9-s + 0.355·10-s + (−0.294 − 0.509i)11-s + (0.316 + 0.187i)12-s + (−0.138 + 0.240i)13-s + (−0.575 + 0.996i)14-s + (0.384 + 0.227i)15-s + (0.248 + 0.429i)16-s + 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.152 - 0.988i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.152 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40970 + 1.64369i\)
\(L(\frac12)\) \(\approx\) \(1.40970 + 1.64369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0186 - 1.73i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.562 - 0.973i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.91 - 3.31i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.975 + 1.68i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 6.58T + 17T^{2} \)
19 \( 1 - 0.304T + 19T^{2} \)
23 \( 1 + (3.00 - 5.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.378 + 0.654i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.95 - 6.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.54T + 37T^{2} \)
41 \( 1 + (-5.74 + 9.94i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.06 + 5.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.90 - 8.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.480T + 53T^{2} \)
59 \( 1 + (-4.91 + 8.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.93 + 8.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.805 + 1.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + (-3.35 - 5.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.46 + 7.72i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.53T + 89T^{2} \)
97 \( 1 + (-0.256 - 0.443i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.84877093756190182339247728177, −10.06332884067747106107153530092, −9.142082019949388258444389798694, −8.367468632438499517744308491872, −7.40316666348571508418545416691, −5.72496818776035274594331295535, −5.62580225111223276241319886021, −4.80674909672903283363906416107, −3.42430174597547042302990662507, −1.85838662342732895960432477160, 1.25262421664218385120880258821, 2.38566385779207632731782644222, 3.51014730298807307162176230708, 4.64686412859684278610194878714, 5.91204334912969278089268573191, 7.30014352070185455458258281560, 7.46837880845990171023589665608, 8.362988245137895936760180530432, 10.07467608380309359535462904416, 10.57305624705994180195533800276

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