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Properties

Label 2-585-9.4-c1-0-42
Degree $2$
Conductor $585$
Sign $0.999 + 0.0103i$
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  + (0.909 + 1.57i)2-s + (1.33 − 1.10i)3-s + (−0.654 + 1.13i)4-s + (0.5 − 0.866i)5-s + (2.95 + 1.09i)6-s + (−1.45 − 2.51i)7-s + 1.25·8-s + (0.551 − 2.94i)9-s + 1.81·10-s + (−1.59 − 2.75i)11-s + (0.382 + 2.23i)12-s + (−0.5 + 0.866i)13-s + (2.64 − 4.58i)14-s + (−0.291 − 1.70i)15-s + (2.45 + 4.24i)16-s − 1.97·17-s + ⋯
L(s)  = 1  + (0.643 + 1.11i)2-s + (0.769 − 0.638i)3-s + (−0.327 + 0.566i)4-s + (0.223 − 0.387i)5-s + (1.20 + 0.446i)6-s + (−0.549 − 0.951i)7-s + 0.444·8-s + (0.183 − 0.982i)9-s + 0.575·10-s + (−0.479 − 0.830i)11-s + (0.110 + 0.645i)12-s + (−0.138 + 0.240i)13-s + (0.706 − 1.22i)14-s + (−0.0753 − 0.440i)15-s + (0.613 + 1.06i)16-s − 0.479·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.999 + 0.0103i$
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: \((2,\ 585,\ (\ :1/2),\ 0.999 + 0.0103i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.64820 - 0.0137016i\)
\(L(\frac12)\) \(\approx\) \(2.64820 - 0.0137016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.33 + 1.10i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.909 - 1.57i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.45 + 2.51i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.97T + 17T^{2} \)
19 \( 1 - 6.13T + 19T^{2} \)
23 \( 1 + (2.79 - 4.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.83 - 3.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.09 - 7.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.77T + 37T^{2} \)
41 \( 1 + (1.03 - 1.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.928 + 1.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.03 - 6.98i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.09T + 53T^{2} \)
59 \( 1 + (2.01 - 3.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.50 - 2.61i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.84 - 4.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.716T + 71T^{2} \)
73 \( 1 - 1.12T + 73T^{2} \)
79 \( 1 + (7.28 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0540 + 0.0936i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.32T + 89T^{2} \)
97 \( 1 + (6.46 + 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.58084485095735777166681853771, −9.616918359106817481290524160998, −8.674300925923844607406316914942, −7.65513338009829680002112497901, −7.19791617764530827504511478486, −6.25480262233199772287736690487, −5.38149021691479456171792435736, −4.13053559251954106868813349579, −3.11750245905528289359668245181, −1.27574430423713040055076244492, 2.26072561939462292488685829541, 2.67439159549935045839317605359, 3.79380896363702945385901070492, 4.77574672296221817175477594332, 5.76861747933268500086659644842, 7.27119309936725337493232061641, 8.161344565453799452891321727636, 9.475321994461621217827075553921, 9.829052972815296109856897820825, 10.68747654982553614531476671097

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