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Properties

Label 2-585-117.103-c1-0-17
Degree $2$
Conductor $585$
Sign $0.911 - 0.411i$
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  + (1.07 − 0.619i)2-s + (−1.48 − 0.896i)3-s + (−0.232 + 0.403i)4-s + (−0.866 − 0.5i)5-s + (−2.14 − 0.0433i)6-s + (−0.166 + 0.0962i)7-s + 3.05i·8-s + (1.39 + 2.65i)9-s − 1.23·10-s + (−0.223 + 0.129i)11-s + (0.706 − 0.388i)12-s + (2.62 + 2.47i)13-s + (−0.119 + 0.206i)14-s + (0.835 + 1.51i)15-s + (1.42 + 2.47i)16-s + 3.42·17-s + ⋯
L(s)  = 1  + (0.758 − 0.437i)2-s + (−0.855 − 0.517i)3-s + (−0.116 + 0.201i)4-s + (−0.387 − 0.223i)5-s + (−0.875 − 0.0177i)6-s + (−0.0629 + 0.0363i)7-s + 1.07i·8-s + (0.464 + 0.885i)9-s − 0.391·10-s + (−0.0674 + 0.0389i)11-s + (0.203 − 0.112i)12-s + (0.726 + 0.686i)13-s + (−0.0318 + 0.0551i)14-s + (0.215 + 0.391i)15-s + (0.356 + 0.617i)16-s + 0.830·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.911 - 0.411i$
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: \((2,\ 585,\ (\ :1/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31824 + 0.283958i\)
\(L(\frac12)\) \(\approx\) \(1.31824 + 0.283958i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.48 + 0.896i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-2.62 - 2.47i)T \)
good2 \( 1 + (-1.07 + 0.619i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.166 - 0.0962i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.223 - 0.129i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 3.42T + 17T^{2} \)
19 \( 1 - 2.08iT - 19T^{2} \)
23 \( 1 + (-0.614 + 1.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.55 - 7.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.13 - 1.23i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.41iT - 37T^{2} \)
41 \( 1 + (9.38 + 5.41i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.21 - 9.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.32 + 1.34i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.93T + 53T^{2} \)
59 \( 1 + (10.3 + 5.96i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.57 - 4.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.23 + 4.75i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.11iT - 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 + (2.92 + 5.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.67 + 2.12i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.03iT - 89T^{2} \)
97 \( 1 + (-5.71 + 3.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.07201154272986333622705005143, −10.26805836173846436406353342991, −8.843631571670821421497775250737, −8.064010313833070378157456852692, −7.08037730477267327311612855085, −6.01118356076450506042068273796, −5.07530669568725155994895637197, −4.24114878996009952337167687381, −3.08823930181981295385927211802, −1.47871127756941759338084591076, 0.74928464940379784950942610905, 3.31694319611160235438545494446, 4.20383054557145836568341362524, 5.15303771118371340325767269844, 5.93463304224928079289635735488, 6.66251894364049738688547883829, 7.74494457414787650391893680354, 9.010392275886328383485156873950, 10.09580398337278648667076466901, 10.50278254331575582778735590775

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