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Properties

Label 2-585-9.7-c1-0-4
Degree $2$
Conductor $585$
Sign $-0.121 + 0.992i$
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  + (−0.780 + 1.35i)2-s + (0.0458 + 1.73i)3-s + (−0.219 − 0.380i)4-s + (−0.5 − 0.866i)5-s + (−2.37 − 1.29i)6-s + (−2.47 + 4.28i)7-s − 2.43·8-s + (−2.99 + 0.158i)9-s + 1.56·10-s + (0.149 − 0.259i)11-s + (0.648 − 0.397i)12-s + (0.5 + 0.866i)13-s + (−3.86 − 6.68i)14-s + (1.47 − 0.905i)15-s + (2.34 − 4.05i)16-s + 8.20·17-s + ⋯
L(s)  = 1  + (−0.552 + 0.956i)2-s + (0.0264 + 0.999i)3-s + (−0.109 − 0.190i)4-s + (−0.223 − 0.387i)5-s + (−0.970 − 0.526i)6-s + (−0.934 + 1.61i)7-s − 0.861·8-s + (−0.998 + 0.0529i)9-s + 0.493·10-s + (0.0451 − 0.0781i)11-s + (0.187 − 0.114i)12-s + (0.138 + 0.240i)13-s + (−1.03 − 1.78i)14-s + (0.381 − 0.233i)15-s + (0.585 − 1.01i)16-s + 1.98·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.121 + 0.992i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.121 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.347171 - 0.392175i\)
\(L(\frac12)\) \(\approx\) \(0.347171 - 0.392175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0458 - 1.73i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.780 - 1.35i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (2.47 - 4.28i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.149 + 0.259i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 8.20T + 17T^{2} \)
19 \( 1 + 3.11T + 19T^{2} \)
23 \( 1 + (0.916 + 1.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.905 - 1.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.503 + 0.872i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.66T + 37T^{2} \)
41 \( 1 + (0.897 + 1.55i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.48 - 9.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.79 + 4.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.636T + 53T^{2} \)
59 \( 1 + (-0.367 - 0.636i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.66 + 2.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.07 - 8.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.479T + 71T^{2} \)
73 \( 1 - 8.42T + 73T^{2} \)
79 \( 1 + (7.32 - 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.26 - 14.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + (7.11 - 12.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.36415326262798922345852246430, −9.915537462410246252906192591827, −9.511934292416570495262249859099, −8.544537517200186313478967576646, −8.249454592541841286314940428830, −6.78042160869149358072599641338, −5.80156421779545123797780104770, −5.32825395740056131523938999628, −3.67263799676139795956438436026, −2.76022643470392563807647022084, 0.35305248792954965504176638427, 1.50722512598878279567892070145, 3.04941616785079294350244831047, 3.70168905857336830161737861662, 5.71367164208300837008318677411, 6.64593578299912190950420348220, 7.39743016606893149426311332113, 8.242374894109151967391058751359, 9.488713928824121862066579823405, 10.28697984511199048700481570523

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