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Properties

Label 2-585-9.7-c1-0-28
Degree $2$
Conductor $585$
Sign $0.850 + 0.526i$
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  + (−0.644 + 1.11i)2-s + (−0.971 − 1.43i)3-s + (0.170 + 0.294i)4-s + (0.5 + 0.866i)5-s + (2.22 − 0.160i)6-s + (1.40 − 2.44i)7-s − 3.01·8-s + (−1.11 + 2.78i)9-s − 1.28·10-s + (2.57 − 4.46i)11-s + (0.257 − 0.530i)12-s + (−0.5 − 0.866i)13-s + (1.81 + 3.14i)14-s + (0.755 − 1.55i)15-s + (1.60 − 2.77i)16-s − 6.95·17-s + ⋯
L(s)  = 1  + (−0.455 + 0.788i)2-s + (−0.561 − 0.827i)3-s + (0.0850 + 0.147i)4-s + (0.223 + 0.387i)5-s + (0.908 − 0.0655i)6-s + (0.532 − 0.922i)7-s − 1.06·8-s + (−0.370 + 0.928i)9-s − 0.407·10-s + (0.776 − 1.34i)11-s + (0.0742 − 0.153i)12-s + (−0.138 − 0.240i)13-s + (0.485 + 0.840i)14-s + (0.195 − 0.402i)15-s + (0.400 − 0.693i)16-s − 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.850 + 0.526i$
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: \((2,\ 585,\ (\ :1/2),\ 0.850 + 0.526i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914432 - 0.260042i\)
\(L(\frac12)\) \(\approx\) \(0.914432 - 0.260042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.971 + 1.43i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.644 - 1.11i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-1.40 + 2.44i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.57 + 4.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 6.95T + 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 + (0.178 + 0.309i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.41 + 4.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.69 + 6.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.70T + 37T^{2} \)
41 \( 1 + (-3.03 - 5.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.89 + 8.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.06 + 5.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.26T + 53T^{2} \)
59 \( 1 + (0.656 + 1.13i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.58 + 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.894 + 1.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.91T + 71T^{2} \)
73 \( 1 - 4.67T + 73T^{2} \)
79 \( 1 + (4.43 - 7.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.47 - 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.44T + 89T^{2} \)
97 \( 1 + (5.62 - 9.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.03676798846217473585231901015, −9.559650185269504736727282955418, −8.534185438334734714896317640573, −7.77697852202721287774427201009, −7.04942149014804331326448983838, −6.34833515761658574024811209510, −5.56729895379020075373252280640, −3.99583390497069853099051569653, −2.52065282075034728974715763963, −0.70814180530090496627514212847, 1.46257627599530556795816206380, 2.65573068967090731065154081721, 4.27043272688679659530558506798, 5.11569384094223243991978300187, 6.05201350803947101587311531433, 7.07192123156918104127678952130, 8.916400655135383393493164394055, 9.093934497094202483312873090347, 9.896042715023808564219443638706, 10.79801314464297601341688312981

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