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Properties

Label 2-585-9.7-c1-0-46
Degree $2$
Conductor $585$
Sign $-0.600 - 0.799i$
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  + (1.21 − 2.09i)2-s + (−0.689 − 1.58i)3-s + (−1.93 − 3.35i)4-s + (−0.5 − 0.866i)5-s + (−4.17 − 0.478i)6-s + (0.905 − 1.56i)7-s − 4.54·8-s + (−2.04 + 2.19i)9-s − 2.42·10-s + (−0.367 + 0.635i)11-s + (−3.99 + 5.39i)12-s + (−0.5 − 0.866i)13-s + (−2.19 − 3.80i)14-s + (−1.03 + 1.39i)15-s + (−1.63 + 2.82i)16-s + 4.77·17-s + ⋯
L(s)  = 1  + (0.856 − 1.48i)2-s + (−0.397 − 0.917i)3-s + (−0.968 − 1.67i)4-s + (−0.223 − 0.387i)5-s + (−1.70 − 0.195i)6-s + (0.342 − 0.592i)7-s − 1.60·8-s + (−0.683 + 0.730i)9-s − 0.766·10-s + (−0.110 + 0.191i)11-s + (−1.15 + 1.55i)12-s + (−0.138 − 0.240i)13-s + (−0.586 − 1.01i)14-s + (−0.266 + 0.359i)15-s + (−0.407 + 0.706i)16-s + 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.600 - 0.799i$
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: \((2,\ 585,\ (\ :1/2),\ -0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.746160 + 1.49340i\)
\(L(\frac12)\) \(\approx\) \(0.746160 + 1.49340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.689 + 1.58i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-1.21 + 2.09i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.905 + 1.56i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.367 - 0.635i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 4.77T + 17T^{2} \)
19 \( 1 + 4.07T + 19T^{2} \)
23 \( 1 + (0.336 + 0.582i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.25 + 2.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.27 - 5.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.08T + 37T^{2} \)
41 \( 1 + (1.56 + 2.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.46 + 7.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.78 - 3.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 + (6.66 + 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.42 + 4.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.43 - 2.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.86T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + (-2.05 + 3.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.35 - 11.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.47T + 89T^{2} \)
97 \( 1 + (-0.851 + 1.47i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60176251566314351261972540080, −9.672885815866695018409703277800, −8.315931692263272946007963886989, −7.49604955079213789843667344939, −6.20387727400462797712535229379, −5.16512696027714980245781438678, −4.38966805167842899797963014337, −3.13982729735380299043298368077, −1.89934266470487007404717207214, −0.78246052598545691982104794144, 3.01802222892111318661480984153, 4.14718630502662259904462718093, 4.89877919024502496337130966809, 5.86434304312239102638973848269, 6.37723176482088602165082502440, 7.63886837784951518007420891175, 8.348355154992968284292123672171, 9.331705534186036000218692809418, 10.37585094053189537421523238374, 11.40473251115887468974553066690

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