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Properties

Label 2-585-9.7-c1-0-41
Degree $2$
Conductor $585$
Sign $-0.226 + 0.974i$
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  + (0.987 − 1.71i)2-s + (1.73 + 0.0465i)3-s + (−0.949 − 1.64i)4-s + (−0.5 − 0.866i)5-s + (1.78 − 2.91i)6-s + (0.124 − 0.215i)7-s + 0.198·8-s + (2.99 + 0.161i)9-s − 1.97·10-s + (−0.0967 + 0.167i)11-s + (−1.56 − 2.89i)12-s + (−0.5 − 0.866i)13-s + (−0.245 − 0.425i)14-s + (−0.825 − 1.52i)15-s + (2.09 − 3.62i)16-s − 0.423·17-s + ⋯
L(s)  = 1  + (0.698 − 1.20i)2-s + (0.999 + 0.0268i)3-s + (−0.474 − 0.822i)4-s + (−0.223 − 0.387i)5-s + (0.730 − 1.19i)6-s + (0.0470 − 0.0814i)7-s + 0.0700·8-s + (0.998 + 0.0537i)9-s − 0.624·10-s + (−0.0291 + 0.0505i)11-s + (−0.452 − 0.835i)12-s + (−0.138 − 0.240i)13-s + (−0.0657 − 0.113i)14-s + (−0.213 − 0.393i)15-s + (0.523 − 0.907i)16-s − 0.102·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.226 + 0.974i$
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: \((2,\ 585,\ (\ :1/2),\ -0.226 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80249 - 2.26936i\)
\(L(\frac12)\) \(\approx\) \(1.80249 - 2.26936i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0465i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.987 + 1.71i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.124 + 0.215i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0967 - 0.167i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.423T + 17T^{2} \)
19 \( 1 + 3.02T + 19T^{2} \)
23 \( 1 + (-1.86 - 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.51 - 4.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.43 + 4.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.39T + 37T^{2} \)
41 \( 1 + (-0.733 - 1.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.485 - 0.841i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.13 - 7.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.91T + 53T^{2} \)
59 \( 1 + (-5.30 - 9.18i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.07 + 10.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.37 - 5.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.47T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + (-2.03 + 3.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.79 - 4.84i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.19T + 89T^{2} \)
97 \( 1 + (-1.75 + 3.03i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.57545183117606451162593646544, −9.715803776810179146250602698256, −8.885934178597314615295784488300, −7.898768628399225349988637763177, −7.05870333026490009192543713277, −5.38694148517730096814950842298, −4.36801326305251461064491823539, −3.61055139111548068256009163761, −2.59270251401363407547948165554, −1.46247097402448756162657487871, 2.11183885807659092785298238530, 3.54860112755432133606905902769, 4.39842652537806134558948285102, 5.47553204157178854585524869887, 6.75313749356156395930770051863, 7.11472231505738723341449148683, 8.210157715701625383572483834323, 8.746146126653766454458861828883, 10.00258951207551617263136825951, 10.80347480739979990753837302912

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