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Properties

Label 2-585-9.4-c1-0-7
Degree $2$
Conductor $585$
Sign $-0.313 + 0.949i$
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  + (−1.26 − 2.18i)2-s + (−1.67 − 0.422i)3-s + (−2.17 + 3.76i)4-s + (0.5 − 0.866i)5-s + (1.19 + 4.19i)6-s + (−1.50 − 2.60i)7-s + 5.92·8-s + (2.64 + 1.42i)9-s − 2.52·10-s + (2.87 + 4.97i)11-s + (5.24 − 5.40i)12-s + (−0.5 + 0.866i)13-s + (−3.79 + 6.57i)14-s + (−1.20 + 1.24i)15-s + (−3.11 − 5.39i)16-s + 5.46·17-s + ⋯
L(s)  = 1  + (−0.890 − 1.54i)2-s + (−0.969 − 0.244i)3-s + (−1.08 + 1.88i)4-s + (0.223 − 0.387i)5-s + (0.487 + 1.71i)6-s + (−0.569 − 0.985i)7-s + 2.09·8-s + (0.880 + 0.473i)9-s − 0.796·10-s + (0.866 + 1.50i)11-s + (1.51 − 1.56i)12-s + (−0.138 + 0.240i)13-s + (−1.01 + 1.75i)14-s + (−0.311 + 0.321i)15-s + (−0.778 − 1.34i)16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.313 + 0.949i$
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: \((2,\ 585,\ (\ :1/2),\ -0.313 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.373262 - 0.516176i\)
\(L(\frac12)\) \(\approx\) \(0.373262 - 0.516176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 + 0.422i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.26 + 2.18i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.50 + 2.60i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.87 - 4.97i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.46T + 17T^{2} \)
19 \( 1 - 5.89T + 19T^{2} \)
23 \( 1 + (4.19 - 7.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.68 + 2.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.29T + 37T^{2} \)
41 \( 1 + (-3.28 + 5.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.327 - 0.566i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.20 - 9.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.03T + 53T^{2} \)
59 \( 1 + (3.77 - 6.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.01 + 5.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.84 - 4.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.00T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + (6.82 + 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.08 - 5.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.25T + 89T^{2} \)
97 \( 1 + (3.12 + 5.41i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.25090414689506193017041046150, −9.780742221707352230088745220930, −9.384123794994402402704779000034, −7.63610261377311292779916882596, −7.29753243736849187417978565031, −5.78003102867592874518719218310, −4.41265439584302069596375486042, −3.61259766313228066271541831391, −1.82481309329494673883741774961, −0.921213819405359057066707414972, 0.843529060437873594366312709754, 3.38566844762219085407470958926, 5.17224434219589032938754109936, 5.86311923809519914754080402657, 6.30222893472220335503516517547, 7.21299142190507723484237271877, 8.335249560713537577575378825283, 9.132296858722939117660854944681, 9.876928267810314198735638260183, 10.57731441663064023581271811233

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