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Properties

Label 2-585-9.4-c1-0-34
Degree $2$
Conductor $585$
Sign $0.669 + 0.743i$
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  + (1.38 + 2.40i)2-s + (−1.66 + 0.477i)3-s + (−2.84 + 4.93i)4-s + (0.5 − 0.866i)5-s + (−3.45 − 3.33i)6-s + (−2.15 − 3.73i)7-s − 10.2·8-s + (2.54 − 1.58i)9-s + 2.77·10-s + (−1.03 − 1.79i)11-s + (2.38 − 9.57i)12-s + (−0.5 + 0.866i)13-s + (5.97 − 10.3i)14-s + (−0.418 + 1.68i)15-s + (−8.51 − 14.7i)16-s − 1.79·17-s + ⋯
L(s)  = 1  + (0.980 + 1.69i)2-s + (−0.961 + 0.275i)3-s + (−1.42 + 2.46i)4-s + (0.223 − 0.387i)5-s + (−1.41 − 1.36i)6-s + (−0.814 − 1.41i)7-s − 3.62·8-s + (0.848 − 0.529i)9-s + 0.877·10-s + (−0.312 − 0.541i)11-s + (0.688 − 2.76i)12-s + (−0.138 + 0.240i)13-s + (1.59 − 2.76i)14-s + (−0.108 + 0.433i)15-s + (−2.12 − 3.68i)16-s − 0.434·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.669 + 0.743i$
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: \((2,\ 585,\ (\ :1/2),\ 0.669 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0589567 - 0.0262468i\)
\(L(\frac12)\) \(\approx\) \(0.0589567 - 0.0262468i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.66 - 0.477i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.38 - 2.40i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (2.15 + 3.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.03 + 1.79i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.79T + 17T^{2} \)
19 \( 1 + 5.17T + 19T^{2} \)
23 \( 1 + (4.10 - 7.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.43 - 2.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.278 + 0.483i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.43T + 37T^{2} \)
41 \( 1 + (-1.14 + 1.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.479 + 0.831i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.36 + 4.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.68T + 53T^{2} \)
59 \( 1 + (-2.06 + 3.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.226 + 0.391i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.14 - 5.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.62T + 71T^{2} \)
73 \( 1 + 4.17T + 73T^{2} \)
79 \( 1 + (-5.99 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.53 + 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (1.18 + 2.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.53869650081934278754179485681, −9.580785320186852974111237984376, −8.546915621532342336774707449590, −7.41577153809680061613670569863, −6.77025936463048106726224352664, −6.04979621193738057179776584986, −5.20415302647098071090271808778, −4.21537602236106509080642551786, −3.61373413243801451797443035791, −0.02788764601782861775888046544, 2.05303549111955619089279770144, 2.68535031897844649697476212033, 4.19006318321256796419718786644, 5.10707790397293033356905229100, 6.04084898653009358409575674294, 6.51316354292533141286012747285, 8.591680247807321695587188044681, 9.636854198119095471342344912911, 10.32364270084817477171330801803, 10.88800211723500369646824021212

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