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Properties

Label 2-585-13.9-c1-0-22
Degree $2$
Conductor $585$
Sign $-0.833 - 0.551i$
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  + (1.31 − 2.27i)2-s + (−2.46 − 4.26i)4-s − 5-s + (−0.544 − 0.943i)7-s − 7.70·8-s + (−1.31 + 2.27i)10-s + (2.36 − 4.08i)11-s + (−3.42 − 1.12i)13-s − 2.86·14-s + (−5.21 + 9.03i)16-s + (2.61 + 4.52i)17-s + (−1.46 − 2.53i)19-s + (2.46 + 4.26i)20-s + (−6.21 − 10.7i)22-s + (−3.85 + 6.67i)23-s + ⋯
L(s)  = 1  + (0.930 − 1.61i)2-s + (−1.23 − 2.13i)4-s − 0.447·5-s + (−0.205 − 0.356i)7-s − 2.72·8-s + (−0.416 + 0.720i)10-s + (0.711 − 1.23i)11-s + (−0.950 − 0.311i)13-s − 0.766·14-s + (−1.30 + 2.25i)16-s + (0.633 + 1.09i)17-s + (−0.335 − 0.581i)19-s + (0.550 + 0.954i)20-s + (−1.32 − 2.29i)22-s + (−0.803 + 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.833 - 0.551i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.833 - 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.469084 + 1.55852i\)
\(L(\frac12)\) \(\approx\) \(0.469084 + 1.55852i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
13 \( 1 + (3.42 + 1.12i)T \)
good2 \( 1 + (-1.31 + 2.27i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (0.544 + 0.943i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.36 + 4.08i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.61 - 4.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.46 + 2.53i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.85 - 6.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.655 + 1.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 + (-5.21 + 9.03i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.49 + 4.31i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.98 + 5.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.51T + 47T^{2} \)
53 \( 1 - 9.67T + 53T^{2} \)
59 \( 1 + (1.58 + 2.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 + 5.47i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.787 + 1.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.00 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + 3.04T + 79T^{2} \)
83 \( 1 - 1.40T + 83T^{2} \)
89 \( 1 + (4.33 - 7.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.51 - 9.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.53903105520803753554889190489, −9.654568946162993343030413200368, −8.798313065236279693180951475969, −7.52689725686641881873765139376, −6.06724684635713395740560549709, −5.30554765851031916692649721412, −3.92077195501973632658157754023, −3.59898451390508016439093050041, −2.22218455976849604029564765924, −0.68718210871169463147620788087, 2.76070713127332083937670663946, 4.22006604614652426237746807935, 4.68524928320936213867480436477, 5.83918712235029284723968454799, 6.77207919258698047626247017530, 7.37593261857841902302011206518, 8.201057074138691571873015402332, 9.205356025104882436292201721910, 10.03386476886474593946524370276, 11.84255003046716754632586519751

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