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Properties

Label 2-585-9.7-c1-0-3
Degree $2$
Conductor $585$
Sign $-0.920 + 0.390i$
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  + (−1.01 + 1.75i)2-s + (0.826 − 1.52i)3-s + (−1.04 − 1.81i)4-s + (0.5 + 0.866i)5-s + (1.83 + 2.98i)6-s + (−1.83 + 3.17i)7-s + 0.196·8-s + (−1.63 − 2.51i)9-s − 2.02·10-s + (−1.32 + 2.29i)11-s + (−3.63 + 0.0946i)12-s + (−0.5 − 0.866i)13-s + (−3.71 − 6.42i)14-s + (1.73 − 0.0451i)15-s + (1.89 − 3.28i)16-s − 5.42·17-s + ⋯
L(s)  = 1  + (−0.715 + 1.23i)2-s + (0.477 − 0.878i)3-s + (−0.524 − 0.908i)4-s + (0.223 + 0.387i)5-s + (0.747 + 1.22i)6-s + (−0.692 + 1.20i)7-s + 0.0696·8-s + (−0.544 − 0.838i)9-s − 0.640·10-s + (−0.399 + 0.691i)11-s + (−1.04 + 0.0273i)12-s + (−0.138 − 0.240i)13-s + (−0.991 − 1.71i)14-s + (0.447 − 0.0116i)15-s + (0.474 − 0.821i)16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.920 + 0.390i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.920 + 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0857915 - 0.421938i\)
\(L(\frac12)\) \(\approx\) \(0.0857915 - 0.421938i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.826 + 1.52i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.01 - 1.75i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (1.83 - 3.17i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.32 - 2.29i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 5.42T + 17T^{2} \)
19 \( 1 - 2.77T + 19T^{2} \)
23 \( 1 + (-0.0274 - 0.0475i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.85 - 8.41i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.73 + 6.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.97T + 37T^{2} \)
41 \( 1 + (1.46 + 2.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.38 - 5.86i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.28 - 3.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.12T + 53T^{2} \)
59 \( 1 + (-5.28 - 9.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.30 - 2.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.219 + 0.380i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.80T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + (-7.95 + 13.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.85 - 4.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.89T + 89T^{2} \)
97 \( 1 + (8.36 - 14.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.17143333214141561915906748829, −9.721338375316918490437695626135, −9.192615217799244179731870280590, −8.499326647450461245208236129932, −7.49718624151691243821229486903, −6.90167525287750116112309477446, −6.11337801845619391634437509944, −5.30880710452026879221056285781, −3.18610357289351204373361813180, −2.15426802332286875193522099276, 0.26519783945984330807737100955, 2.07226988866949521046451325086, 3.31774400513003949332858645641, 4.00495579545415496567321630658, 5.29310444197801555671285797264, 6.70058917672623067250380864809, 8.031370579833525736643291409134, 8.841910131979954750659014593749, 9.601180316451910764998813523078, 10.11874818324146202876288485312

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