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Properties

Label 2-585-117.103-c1-0-18
Degree $2$
Conductor $585$
Sign $-0.638 - 0.769i$
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  + (−1.22 + 0.707i)2-s + (1.65 + 0.501i)3-s + (0.00235 − 0.00407i)4-s + (0.866 + 0.5i)5-s + (−2.38 + 0.559i)6-s + (−0.00992 + 0.00573i)7-s − 2.82i·8-s + (2.49 + 1.66i)9-s − 1.41·10-s + (−3.20 + 1.85i)11-s + (0.00593 − 0.00557i)12-s + (−3.01 + 1.98i)13-s + (0.00811 − 0.0140i)14-s + (1.18 + 1.26i)15-s + (2.00 + 3.47i)16-s + 4.64·17-s + ⋯
L(s)  = 1  + (−0.867 + 0.500i)2-s + (0.957 + 0.289i)3-s + (0.00117 − 0.00203i)4-s + (0.387 + 0.223i)5-s + (−0.974 + 0.228i)6-s + (−0.00375 + 0.00216i)7-s − 0.998i·8-s + (0.832 + 0.553i)9-s − 0.447·10-s + (−0.967 + 0.558i)11-s + (0.00171 − 0.00160i)12-s + (−0.835 + 0.549i)13-s + (0.00216 − 0.00375i)14-s + (0.306 + 0.326i)15-s + (0.501 + 0.868i)16-s + 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.638 - 0.769i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477078 + 1.01604i\)
\(L(\frac12)\) \(\approx\) \(0.477078 + 1.01604i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 - 0.501i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (3.01 - 1.98i)T \)
good2 \( 1 + (1.22 - 0.707i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (0.00992 - 0.00573i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.20 - 1.85i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.64T + 17T^{2} \)
19 \( 1 - 6.04iT - 19T^{2} \)
23 \( 1 + (2.42 - 4.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.46 + 7.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.72 - 3.30i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.52iT - 37T^{2} \)
41 \( 1 + (3.46 + 1.99i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.571 + 0.989i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-8.97 + 5.17i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.37T + 53T^{2} \)
59 \( 1 + (1.19 + 0.688i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.33 + 7.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.37 - 4.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.79iT - 71T^{2} \)
73 \( 1 - 1.97iT - 73T^{2} \)
79 \( 1 + (0.376 + 0.652i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.33 - 4.23i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.56iT - 89T^{2} \)
97 \( 1 + (0.00405 - 0.00234i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.29137158935382138887211789764, −9.946030141425336462069706505375, −9.384646989073231320024507180132, −8.158732353650520220615655770369, −7.78496491523101005057966551147, −6.98376863928118016853990924316, −5.62414949856915617120619183268, −4.31143060750994924165573486761, −3.21647592620023495895702658283, −1.88075552708581127238296673892, 0.75789377070979368005790261957, 2.26256473562291649521183802016, 3.03449431576945645834454540774, 4.79278180005415926676066151433, 5.72733681008471071701672093386, 7.21355606119205485260799151661, 8.038032894791203984115587058148, 8.709434856432866525125845056902, 9.528964357206655284192359732244, 10.17817489898425316103988780211

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