Properties

Label 2-585-13.4-c1-0-8
Degree $2$
Conductor $585$
Sign $0.515 + 0.856i$
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  + (−1.05 − 0.609i)2-s + (−0.256 − 0.443i)4-s i·5-s + (3.11 − 1.80i)7-s + 3.06i·8-s + (−0.609 + 1.05i)10-s + (4.65 + 2.68i)11-s + (1.81 + 3.11i)13-s − 4.39·14-s + (1.35 − 2.34i)16-s + (0.565 + 0.980i)17-s + (−1.96 + 1.13i)19-s + (−0.443 + 0.256i)20-s + (−3.27 − 5.67i)22-s + (1.94 − 3.37i)23-s + ⋯
L(s)  = 1  + (−0.746 − 0.431i)2-s + (−0.128 − 0.221i)4-s − 0.447i·5-s + (1.17 − 0.680i)7-s + 1.08i·8-s + (−0.192 + 0.334i)10-s + (1.40 + 0.809i)11-s + (0.504 + 0.863i)13-s − 1.17·14-s + (0.339 − 0.587i)16-s + (0.137 + 0.237i)17-s + (−0.450 + 0.260i)19-s + (−0.0992 + 0.0572i)20-s + (−0.698 − 1.20i)22-s + (0.405 − 0.702i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.515 + 0.856i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00042 - 0.565447i\)
\(L(\frac12)\) \(\approx\) \(1.00042 - 0.565447i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + iT \)
13 \( 1 + (-1.81 - 3.11i)T \)
good2 \( 1 + (1.05 + 0.609i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-3.11 + 1.80i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.65 - 2.68i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.565 - 0.980i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.96 - 1.13i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.94 + 3.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0123 - 0.0214i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + (-7.53 - 4.35i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.23 + 1.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.565 + 0.980i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.58iT - 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + (-0.148 + 0.0857i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.68 + 2.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.54 - 3.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.35 - 5.39i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.70iT - 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 + (13.9 + 8.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.5 - 6.08i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.47037313948120179358462678912, −9.679195642188487370880048480268, −8.851911542890569506097048168131, −8.246390820032673773223738726590, −7.15066189096798653861878306373, −6.03021035033725017364334153258, −4.66027143459471635066458444635, −4.16208190302162688224111251859, −1.93934773045573340028326406107, −1.15885647089443219427386733001, 1.23550543909298884765894061055, 3.07043982705124018575311043362, 4.16340033708136951503235893315, 5.52829986860158000082065188189, 6.49310403351457170257459829511, 7.46854934902979942473585719179, 8.377989568207921333825491993155, 8.816240910419103443217280835452, 9.720346077584528605635621661742, 10.91460215919695323220156172602

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