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Properties

Label 2-585-9.4-c1-0-22
Degree $2$
Conductor $585$
Sign $0.821 - 0.570i$
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  + (0.628 + 1.08i)2-s + (−0.933 + 1.45i)3-s + (0.209 − 0.363i)4-s + (0.5 − 0.866i)5-s + (−2.17 − 0.0995i)6-s + (−2.28 − 3.95i)7-s + 3.04·8-s + (−1.25 − 2.72i)9-s + 1.25·10-s + (2.59 + 4.49i)11-s + (0.333 + 0.644i)12-s + (−0.5 + 0.866i)13-s + (2.87 − 4.97i)14-s + (0.796 + 1.53i)15-s + (1.49 + 2.58i)16-s + 4.28·17-s + ⋯
L(s)  = 1  + (0.444 + 0.769i)2-s + (−0.539 + 0.842i)3-s + (0.104 − 0.181i)4-s + (0.223 − 0.387i)5-s + (−0.888 − 0.0406i)6-s + (−0.863 − 1.49i)7-s + 1.07·8-s + (−0.418 − 0.908i)9-s + 0.397·10-s + (0.781 + 1.35i)11-s + (0.0964 + 0.186i)12-s + (−0.138 + 0.240i)13-s + (0.767 − 1.32i)14-s + (0.205 + 0.397i)15-s + (0.373 + 0.646i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.821 - 0.570i$
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: \((2,\ 585,\ (\ :1/2),\ 0.821 - 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67294 + 0.523652i\)
\(L(\frac12)\) \(\approx\) \(1.67294 + 0.523652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.933 - 1.45i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.628 - 1.08i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (2.28 + 3.95i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.59 - 4.49i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.28T + 17T^{2} \)
19 \( 1 - 4.84T + 19T^{2} \)
23 \( 1 + (-3.78 + 6.55i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.109 - 0.189i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.619 + 1.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.15T + 37T^{2} \)
41 \( 1 + (5.98 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.828 - 1.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.53 + 9.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.54T + 53T^{2} \)
59 \( 1 + (-2.00 + 3.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.97 + 6.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.78 + 3.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + 9.80T + 73T^{2} \)
79 \( 1 + (-1.68 - 2.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.12 + 3.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.708T + 89T^{2} \)
97 \( 1 + (-1.78 - 3.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.52125163695344749148619875209, −9.874014519927218630296043734988, −9.496288747073445984666664429517, −7.80812752617182777852100404376, −6.78479892639741817773066855388, −6.43484525192729427377995815674, −5.05650864294379362472081506665, −4.53424625979503550598418042709, −3.50605008934240935409820579885, −1.11757043596710386622440427230, 1.43149606808877758947994703238, 2.89613865224147361592147755073, 3.31871609371121995326735874541, 5.36570632151166987546508241176, 5.87365576062403520475295184574, 6.90783969872109937265603193554, 7.85176024371153322874577938362, 8.931069000529797694278650347687, 9.899457882528980565812281083656, 11.07639577568041518965450898462

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