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Properties

Label 2-585-117.103-c1-0-3
Degree $2$
Conductor $585$
Sign $-0.489 + 0.872i$
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  + (−0.0488 + 0.0282i)2-s + (−1.54 + 0.787i)3-s + (−0.998 + 1.72i)4-s + (0.866 + 0.5i)5-s + (0.0531 − 0.0819i)6-s + (−3.39 + 1.96i)7-s − 0.225i·8-s + (1.76 − 2.42i)9-s − 0.0564·10-s + (−1.93 + 1.11i)11-s + (0.179 − 3.45i)12-s + (−1.98 + 3.01i)13-s + (0.110 − 0.191i)14-s + (−1.72 − 0.0896i)15-s + (−1.99 − 3.44i)16-s + 5.60·17-s + ⋯
L(s)  = 1  + (−0.0345 + 0.0199i)2-s + (−0.890 + 0.454i)3-s + (−0.499 + 0.864i)4-s + (0.387 + 0.223i)5-s + (0.0217 − 0.0334i)6-s + (−1.28 + 0.741i)7-s − 0.0797i·8-s + (0.586 − 0.809i)9-s − 0.0178·10-s + (−0.583 + 0.337i)11-s + (0.0516 − 0.997i)12-s + (−0.549 + 0.835i)13-s + (0.0295 − 0.0512i)14-s + (−0.446 − 0.0231i)15-s + (−0.497 − 0.861i)16-s + 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.489 + 0.872i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.489 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0681098 - 0.116315i\)
\(L(\frac12)\) \(\approx\) \(0.0681098 - 0.116315i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.54 - 0.787i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (1.98 - 3.01i)T \)
good2 \( 1 + (0.0488 - 0.0282i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (3.39 - 1.96i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.93 - 1.11i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.60T + 17T^{2} \)
19 \( 1 + 6.06iT - 19T^{2} \)
23 \( 1 + (1.24 - 2.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.26 + 3.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.13 - 0.654i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.55iT - 37T^{2} \)
41 \( 1 + (9.94 + 5.74i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.04 - 7.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.02 + 0.591i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.19T + 53T^{2} \)
59 \( 1 + (3.81 + 2.20i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.24 + 2.15i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.7 + 6.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.1iT - 71T^{2} \)
73 \( 1 - 2.60iT - 73T^{2} \)
79 \( 1 + (1.96 + 3.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.81 - 5.66i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.84iT - 89T^{2} \)
97 \( 1 + (-7.85 + 4.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.42921669924805807650613836027, −10.19277339289299253723126980839, −9.516345830060005188688224759022, −9.068186731216754508124663218967, −7.55798046876773219753214178482, −6.76585659909839740076138024981, −5.75000486659261600218916462542, −4.87708287305319123682028938354, −3.68816405005009986766069280270, −2.63932900342659164392352315656, 0.090636115638353623183688905617, 1.36465451144040986958123275545, 3.27770496960561201068595583986, 4.75576252137031380942484198233, 5.69542289751597222019017014688, 6.16399046688210380578353526899, 7.29645391541189290435130273010, 8.263148041495531306004828882431, 9.703942871642760892848154134495, 10.27607180949257431352350584860

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