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Properties

Label 2-585-117.25-c1-0-54
Degree $2$
Conductor $585$
Sign $-0.456 + 0.889i$
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  + (1.39 + 0.807i)2-s + (0.0517 − 1.73i)3-s + (0.304 + 0.527i)4-s + (−0.866 + 0.5i)5-s + (1.47 − 2.38i)6-s + (−1.67 − 0.966i)7-s − 2.24i·8-s + (−2.99 − 0.179i)9-s − 1.61·10-s + (−5.50 − 3.18i)11-s + (0.929 − 0.500i)12-s + (−3.46 + 1.00i)13-s + (−1.56 − 2.70i)14-s + (0.820 + 1.52i)15-s + (2.42 − 4.19i)16-s + 7.37·17-s + ⋯
L(s)  = 1  + (0.989 + 0.571i)2-s + (0.0298 − 0.999i)3-s + (0.152 + 0.263i)4-s + (−0.387 + 0.223i)5-s + (0.600 − 0.971i)6-s + (−0.632 − 0.365i)7-s − 0.794i·8-s + (−0.998 − 0.0596i)9-s − 0.510·10-s + (−1.66 − 0.959i)11-s + (0.268 − 0.144i)12-s + (−0.960 + 0.279i)13-s + (−0.417 − 0.722i)14-s + (0.211 + 0.393i)15-s + (0.605 − 1.04i)16-s + 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.456 + 0.889i$
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: \((2,\ 585,\ (\ :1/2),\ -0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.708793 - 1.16040i\)
\(L(\frac12)\) \(\approx\) \(0.708793 - 1.16040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0517 + 1.73i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (3.46 - 1.00i)T \)
good2 \( 1 + (-1.39 - 0.807i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (1.67 + 0.966i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.50 + 3.18i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 7.37T + 17T^{2} \)
19 \( 1 - 0.941iT - 19T^{2} \)
23 \( 1 + (-1.21 - 2.11i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.01 - 1.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.79 + 5.07i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.18iT - 37T^{2} \)
41 \( 1 + (-3.06 + 1.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.69 + 8.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.88 + 3.97i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.73T + 53T^{2} \)
59 \( 1 + (-7.79 + 4.50i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.789 + 1.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.551 - 0.318i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + 0.468iT - 73T^{2} \)
79 \( 1 + (-1.56 + 2.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.12 + 1.80i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.3iT - 89T^{2} \)
97 \( 1 + (-6.43 - 3.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.42502888957468319159601688166, −9.663088045147536455903074418071, −8.109118823786590453456706502652, −7.56859141409228438825235812970, −6.75798882294860241288839868907, −5.74988021603274173428372453583, −5.18092208216606617840517697792, −3.60362405435991557243311088786, −2.77279042822773551257342839866, −0.53088247033585145555059887691, 2.73627859562871313911016335482, 3.11693300760210532099577329063, 4.61954907803424470059549935524, 4.95108889963296454980844507942, 5.92848752468458630880533210715, 7.67292122047209550880433945535, 8.269719635273190467961651018690, 9.665773512313712870645944892467, 10.11042232327677296523936765449, 11.03358392750559390775900571688

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