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Properties

Label 2-585-9.7-c1-0-8
Degree $2$
Conductor $585$
Sign $-0.209 - 0.977i$
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  + (−0.0336 + 0.0581i)2-s + (−0.332 − 1.69i)3-s + (0.997 + 1.72i)4-s + (0.5 + 0.866i)5-s + (0.110 + 0.0377i)6-s + (−1.23 + 2.13i)7-s − 0.268·8-s + (−2.77 + 1.12i)9-s − 0.0672·10-s + (−1.60 + 2.78i)11-s + (2.60 − 2.27i)12-s + (0.5 + 0.866i)13-s + (−0.0827 − 0.143i)14-s + (1.30 − 1.13i)15-s + (−1.98 + 3.44i)16-s − 4.77·17-s + ⋯
L(s)  = 1  + (−0.0237 + 0.0411i)2-s + (−0.191 − 0.981i)3-s + (0.498 + 0.864i)4-s + (0.223 + 0.387i)5-s + (0.0449 + 0.0154i)6-s + (−0.465 + 0.806i)7-s − 0.0949·8-s + (−0.926 + 0.376i)9-s − 0.0212·10-s + (−0.484 + 0.838i)11-s + (0.752 − 0.655i)12-s + (0.138 + 0.240i)13-s + (−0.0221 − 0.0383i)14-s + (0.337 − 0.293i)15-s + (−0.496 + 0.860i)16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.209 - 0.977i$
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: \((2,\ 585,\ (\ :1/2),\ -0.209 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.634187 + 0.784717i\)
\(L(\frac12)\) \(\approx\) \(0.634187 + 0.784717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.332 + 1.69i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.0336 - 0.0581i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (1.23 - 2.13i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.60 - 2.78i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 + 3.94T + 19T^{2} \)
23 \( 1 + (2.13 + 3.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.15 + 1.99i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.81 - 6.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.87T + 37T^{2} \)
41 \( 1 + (-1.26 - 2.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.28 + 3.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.13 + 5.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + (1.42 + 2.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.35 - 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.75 - 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.79T + 71T^{2} \)
73 \( 1 - 2.26T + 73T^{2} \)
79 \( 1 + (4.56 - 7.90i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.25 - 9.10i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.966T + 89T^{2} \)
97 \( 1 + (-9.39 + 16.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.13093819599414596763091683091, −10.23542998290748459719618544833, −8.839297034943367550993861600390, −8.319156337541382160883055095778, −7.14397613038449981578901693578, −6.65776203889357554623123803482, −5.81975176714496615118616647738, −4.30942355219764239613200676808, −2.64488251671974312239635788018, −2.22773596093241142507562066238, 0.53712341617138170740605114861, 2.49286484786563013472845096939, 3.87171441222765995041660936645, 4.86036161711703322420365699253, 5.91382987108890554968625566189, 6.46256005809787529842325041211, 7.88572282411615882681477975447, 9.016413816190810513956652021558, 9.703881421955423236325527018827, 10.60498641267837012594965154061

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