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Properties

Label 2-585-9.4-c1-0-5
Degree $2$
Conductor $585$
Sign $0.944 - 0.327i$
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  + (−0.947 − 1.64i)2-s + (1.31 + 1.12i)3-s + (−0.794 + 1.37i)4-s + (−0.5 + 0.866i)5-s + (0.593 − 3.22i)6-s + (−0.837 − 1.45i)7-s − 0.780·8-s + (0.476 + 2.96i)9-s + 1.89·10-s + (3.21 + 5.56i)11-s + (−2.59 + 0.921i)12-s + (−0.5 + 0.866i)13-s + (−1.58 + 2.74i)14-s + (−1.63 + 0.580i)15-s + (2.32 + 4.03i)16-s + 1.05·17-s + ⋯
L(s)  = 1  + (−0.669 − 1.15i)2-s + (0.761 + 0.648i)3-s + (−0.397 + 0.687i)4-s + (−0.223 + 0.387i)5-s + (0.242 − 1.31i)6-s + (−0.316 − 0.548i)7-s − 0.275·8-s + (0.158 + 0.987i)9-s + 0.599·10-s + (0.968 + 1.67i)11-s + (−0.748 + 0.265i)12-s + (−0.138 + 0.240i)13-s + (−0.424 + 0.734i)14-s + (−0.421 + 0.149i)15-s + (0.581 + 1.00i)16-s + 0.256·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.944 - 0.327i$
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: \((2,\ 585,\ (\ :1/2),\ 0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10866 + 0.186951i\)
\(L(\frac12)\) \(\approx\) \(1.10866 + 0.186951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.31 - 1.12i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.947 + 1.64i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.837 + 1.45i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.21 - 5.56i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.05T + 17T^{2} \)
19 \( 1 + 3.65T + 19T^{2} \)
23 \( 1 + (0.680 - 1.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.77 - 6.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.43 + 5.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.00T + 37T^{2} \)
41 \( 1 + (4.24 - 7.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.06 - 7.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.19 - 2.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.53T + 53T^{2} \)
59 \( 1 + (1.08 - 1.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.43 + 5.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.61 + 13.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.39T + 71T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 + (-8.52 - 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.10 + 10.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.02T + 89T^{2} \)
97 \( 1 + (-0.432 - 0.748i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.49691970277044793264892516463, −9.872042096152688147447290707531, −9.421271937065407732111747843825, −8.456829318863860769487377318195, −7.41582078605898274944205213899, −6.45210723866894849941078411533, −4.57624085060982604473303491634, −3.84056049696168040139026527149, −2.74728019912266350864117378003, −1.66027798782257428105105098378, 0.75113314085972478079123530874, 2.73033770458486384152356992801, 3.86229042955087792762196923200, 5.69869208373239728397470948569, 6.31548864124617436551344703166, 7.14989802946702672743848156622, 8.233104040637712333162528773568, 8.664474631767579681577988701277, 9.126067946068292528160231036714, 10.32002425539467209568990324551

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