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Properties

Label 2-585-9.7-c1-0-6
Degree $2$
Conductor $585$
Sign $-0.996 - 0.0860i$
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  + (−0.0377 + 0.0654i)2-s + (−1.16 + 1.27i)3-s + (0.997 + 1.72i)4-s + (−0.5 − 0.866i)5-s + (−0.0394 − 0.124i)6-s + (−0.0159 + 0.0275i)7-s − 0.301·8-s + (−0.264 − 2.98i)9-s + 0.0755·10-s + (−2.85 + 4.95i)11-s + (−3.37 − 0.745i)12-s + (0.5 + 0.866i)13-s + (−0.00120 − 0.00208i)14-s + (1.69 + 0.373i)15-s + (−1.98 + 3.43i)16-s − 3.52·17-s + ⋯
L(s)  = 1  + (−0.0267 + 0.0462i)2-s + (−0.675 + 0.737i)3-s + (0.498 + 0.863i)4-s + (−0.223 − 0.387i)5-s + (−0.0160 − 0.0509i)6-s + (−0.00601 + 0.0104i)7-s − 0.106·8-s + (−0.0882 − 0.996i)9-s + 0.0238·10-s + (−0.862 + 1.49i)11-s + (−0.973 − 0.215i)12-s + (0.138 + 0.240i)13-s + (−0.000321 − 0.000556i)14-s + (0.436 + 0.0965i)15-s + (−0.495 + 0.858i)16-s − 0.853·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.996 - 0.0860i$
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: \((2,\ 585,\ (\ :1/2),\ -0.996 - 0.0860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0291860 + 0.676937i\)
\(L(\frac12)\) \(\approx\) \(0.0291860 + 0.676937i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.16 - 1.27i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.0377 - 0.0654i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (0.0159 - 0.0275i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.85 - 4.95i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.52T + 17T^{2} \)
19 \( 1 - 2.66T + 19T^{2} \)
23 \( 1 + (2.42 + 4.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.23 - 7.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.66 + 2.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.25T + 37T^{2} \)
41 \( 1 + (2.89 + 5.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.83 + 6.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.12 + 1.94i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 9.14T + 53T^{2} \)
59 \( 1 + (-4.02 - 6.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.20 - 5.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.46 - 7.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 2.96T + 73T^{2} \)
79 \( 1 + (3.95 - 6.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.12 + 1.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + (7.48 - 12.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.03177006425038152040417770248, −10.45197187726993601248604615376, −9.363031326537721311921753213261, −8.591958263903520973158867156769, −7.40600867768187572934115599310, −6.80437754363076392630102773238, −5.46959496365685041686848192156, −4.54860555153383314308202508719, −3.68338623347731248490623658707, −2.20877466588644507053650933263, 0.38848297816262599569159619317, 1.94699605183496435087135195002, 3.22071990645100858541095505120, 5.05043489656622307564674528550, 5.83720027674846299463296751244, 6.47489453489037415386238000426, 7.50817211687985326968334949887, 8.263016330888216272478686392752, 9.620543266473097042429577017350, 10.60488688567248544601487058210

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