Properties

Label 2-585-117.25-c1-0-43
Degree $2$
Conductor $585$
Sign $-0.851 + 0.524i$
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  + (0.146 + 0.0844i)2-s + (−0.106 + 1.72i)3-s + (−0.985 − 1.70i)4-s + (−0.866 + 0.5i)5-s + (−0.161 + 0.243i)6-s + (1.65 + 0.957i)7-s − 0.670i·8-s + (−2.97 − 0.368i)9-s − 0.168·10-s + (−3.52 − 2.03i)11-s + (3.05 − 1.52i)12-s + (−3.53 − 0.705i)13-s + (0.161 + 0.280i)14-s + (−0.772 − 1.55i)15-s + (−1.91 + 3.31i)16-s − 0.709·17-s + ⋯
L(s)  = 1  + (0.103 + 0.0597i)2-s + (−0.0615 + 0.998i)3-s + (−0.492 − 0.853i)4-s + (−0.387 + 0.223i)5-s + (−0.0659 + 0.0995i)6-s + (0.627 + 0.362i)7-s − 0.237i·8-s + (−0.992 − 0.122i)9-s − 0.0534·10-s + (−1.06 − 0.613i)11-s + (0.882 − 0.439i)12-s + (−0.980 − 0.195i)13-s + (0.0432 + 0.0749i)14-s + (−0.199 − 0.400i)15-s + (−0.478 + 0.829i)16-s − 0.172·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.851 + 0.524i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.851 + 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00789879 - 0.0278603i\)
\(L(\frac12)\) \(\approx\) \(0.00789879 - 0.0278603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.106 - 1.72i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (3.53 + 0.705i)T \)
good2 \( 1 + (-0.146 - 0.0844i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-1.65 - 0.957i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.52 + 2.03i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.709T + 17T^{2} \)
19 \( 1 - 6.45iT - 19T^{2} \)
23 \( 1 + (4.18 + 7.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.53 + 6.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.30 - 2.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.73iT - 37T^{2} \)
41 \( 1 + (9.47 - 5.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.35 - 7.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.97 + 1.71i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.83T + 53T^{2} \)
59 \( 1 + (-8.61 + 4.97i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.50 + 2.60i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.37 + 1.94i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.70iT - 71T^{2} \)
73 \( 1 - 5.79iT - 73T^{2} \)
79 \( 1 + (6.92 - 11.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.02 + 1.74i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 + (-14.5 - 8.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.16539644448192331939329373722, −9.858081493437164501408993012953, −8.419460296618946010165050246259, −8.137231541985996055679181060724, −6.40819019854695552521411366849, −5.44110255952739452360058572066, −4.85165876834357389106974943280, −3.82596901398007041749183466026, −2.39018285261376082422673454356, −0.01495438965487508714623245306, 2.03358500953094304963120734046, 3.23401730837808672017498211192, 4.68814418244470973832736587451, 5.26688492860473333704619752716, 7.12832577696828291869399080129, 7.37855406860843964056376395040, 8.253795676156376606773202675845, 9.016690655423417466423154531695, 10.24366684907617952278018683224, 11.48253522438579513239436383154

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