Properties

Label 2-585-117.25-c1-0-23
Degree $2$
Conductor $585$
Sign $0.624 + 0.781i$
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  + (−1.09 − 0.634i)2-s + (1.58 − 0.700i)3-s + (−0.194 − 0.337i)4-s + (−0.866 + 0.5i)5-s + (−2.18 − 0.234i)6-s + (4.16 + 2.40i)7-s + 3.03i·8-s + (2.01 − 2.22i)9-s + 1.26·10-s + (−3.25 − 1.88i)11-s + (−0.545 − 0.397i)12-s + (3.09 − 1.85i)13-s + (−3.04 − 5.28i)14-s + (−1.02 + 1.39i)15-s + (1.53 − 2.65i)16-s + 2.36·17-s + ⋯
L(s)  = 1  + (−0.777 − 0.448i)2-s + (0.914 − 0.404i)3-s + (−0.0974 − 0.168i)4-s + (−0.387 + 0.223i)5-s + (−0.892 − 0.0958i)6-s + (1.57 + 0.908i)7-s + 1.07i·8-s + (0.672 − 0.740i)9-s + 0.401·10-s + (−0.982 − 0.566i)11-s + (−0.157 − 0.114i)12-s + (0.858 − 0.513i)13-s + (−0.815 − 1.41i)14-s + (−0.263 + 0.361i)15-s + (0.383 − 0.664i)16-s + 0.572·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.624 + 0.781i$
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: \((2,\ 585,\ (\ :1/2),\ 0.624 + 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26580 - 0.608751i\)
\(L(\frac12)\) \(\approx\) \(1.26580 - 0.608751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.58 + 0.700i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-3.09 + 1.85i)T \)
good2 \( 1 + (1.09 + 0.634i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-4.16 - 2.40i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.25 + 1.88i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.36T + 17T^{2} \)
19 \( 1 - 4.60iT - 19T^{2} \)
23 \( 1 + (-4.07 - 7.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.72 + 4.72i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.31 - 0.756i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.70iT - 37T^{2} \)
41 \( 1 + (-3.95 + 2.28i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.981 - 1.69i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.48 + 4.89i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.05T + 53T^{2} \)
59 \( 1 + (0.839 - 0.484i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.76 + 8.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.76 - 1.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.78iT - 71T^{2} \)
73 \( 1 + 1.99iT - 73T^{2} \)
79 \( 1 + (3.61 - 6.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.04 + 1.75i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.2iT - 89T^{2} \)
97 \( 1 + (2.99 + 1.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.63168036169244347623604601716, −9.543458625721087010876420658865, −8.701931342310324421609310202046, −7.910677176840745021631588250398, −7.85199107264057582607264494583, −5.86976541189051419831518895844, −5.10880872553820193501982016700, −3.49367042222112343199042105407, −2.29448567333412523826339275853, −1.26216863718585636745249912273, 1.30514963175202502408380672797, 3.07398082953193088928919089235, 4.46699893843248344838249883525, 4.74214757591862120699721969873, 6.93229801828585950977894019127, 7.54308970464434492549558200590, 8.406182690366082830352733317765, 8.600526017261435181233524743011, 9.823888254432741441432141276576, 10.61800400772510556312755450191

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