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Properties

Label 2-585-9.4-c1-0-33
Degree $2$
Conductor $585$
Sign $-0.997 + 0.0704i$
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  + (−0.237 − 0.411i)2-s + (−1.28 + 1.15i)3-s + (0.887 − 1.53i)4-s + (−0.5 + 0.866i)5-s + (0.782 + 0.254i)6-s + (−1.24 − 2.16i)7-s − 1.79·8-s + (0.311 − 2.98i)9-s + 0.475·10-s + (1.50 + 2.60i)11-s + (0.639 + 3.00i)12-s + (−0.5 + 0.866i)13-s + (−0.592 + 1.02i)14-s + (−0.360 − 1.69i)15-s + (−1.34 − 2.33i)16-s − 1.86·17-s + ⋯
L(s)  = 1  + (−0.168 − 0.291i)2-s + (−0.742 + 0.669i)3-s + (0.443 − 0.768i)4-s + (−0.223 + 0.387i)5-s + (0.319 + 0.103i)6-s + (−0.471 − 0.816i)7-s − 0.634·8-s + (0.103 − 0.994i)9-s + 0.150·10-s + (0.453 + 0.784i)11-s + (0.184 + 0.867i)12-s + (−0.138 + 0.240i)13-s + (−0.158 + 0.274i)14-s + (−0.0931 − 0.437i)15-s + (−0.336 − 0.583i)16-s − 0.451·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.997 + 0.0704i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.997 + 0.0704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00707505 - 0.200719i\)
\(L(\frac12)\) \(\approx\) \(0.00707505 - 0.200719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 - 1.15i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.237 + 0.411i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.24 + 2.16i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.50 - 2.60i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.86T + 17T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 + (2.00 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.77 + 6.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.552 + 0.957i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + (-1.96 + 3.40i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.18 + 5.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0661 - 0.114i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.91T + 53T^{2} \)
59 \( 1 + (0.402 - 0.697i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.39 - 4.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.07 - 3.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.44T + 71T^{2} \)
73 \( 1 + 6.33T + 73T^{2} \)
79 \( 1 + (-2.31 - 4.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.26 - 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.77T + 89T^{2} \)
97 \( 1 + (3.03 + 5.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.27592646728440456823868334349, −9.823906109920376867653052571353, −8.881456861136230215053049378306, −7.20522170333701486269833776809, −6.62931516961799865187484284223, −5.78076680647040702967206876986, −4.50236772051139411738155440384, −3.67769071785661071143213435310, −1.97474153191801544114127374079, −0.12009540566764818605671167025, 2.03147740279764700657821500032, 3.32961623967615780771766182303, 4.76884679840309792198014628596, 6.08383912630040023257698157398, 6.47070681967951386151899746576, 7.52585183144962185896825753571, 8.523152536997474338027119669492, 8.925267046526079217465710861120, 10.51461000287593931232732970780, 11.29922225449579322517596363704

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