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Properties

Label 2-585-117.25-c1-0-48
Degree $2$
Conductor $585$
Sign $-0.396 + 0.918i$
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  + (0.355 + 0.205i)2-s + (0.841 − 1.51i)3-s + (−0.915 − 1.58i)4-s + (0.866 − 0.5i)5-s + (0.609 − 0.364i)6-s + (0.661 + 0.381i)7-s − 1.57i·8-s + (−1.58 − 2.54i)9-s + 0.410·10-s + (−0.384 − 0.222i)11-s + (−3.17 + 0.0508i)12-s + (0.944 − 3.47i)13-s + (0.156 + 0.271i)14-s + (−0.0277 − 1.73i)15-s + (−1.50 + 2.61i)16-s − 1.99·17-s + ⋯
L(s)  = 1  + (0.251 + 0.144i)2-s + (0.486 − 0.873i)3-s + (−0.457 − 0.793i)4-s + (0.387 − 0.223i)5-s + (0.248 − 0.148i)6-s + (0.250 + 0.144i)7-s − 0.555i·8-s + (−0.527 − 0.849i)9-s + 0.129·10-s + (−0.115 − 0.0669i)11-s + (−0.915 + 0.0146i)12-s + (0.261 − 0.965i)13-s + (0.0418 + 0.0725i)14-s + (−0.00716 − 0.447i)15-s + (−0.377 + 0.653i)16-s − 0.483·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.396 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961937 - 1.46293i\)
\(L(\frac12)\) \(\approx\) \(0.961937 - 1.46293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.841 + 1.51i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.944 + 3.47i)T \)
good2 \( 1 + (-0.355 - 0.205i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-0.661 - 0.381i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.384 + 0.222i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.99T + 17T^{2} \)
19 \( 1 - 3.58iT - 19T^{2} \)
23 \( 1 + (-0.803 - 1.39i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.298 + 0.516i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.94 + 4.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.218iT - 37T^{2} \)
41 \( 1 + (-0.971 + 0.561i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.412 - 0.714i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.55 + 2.05i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.08T + 53T^{2} \)
59 \( 1 + (3.52 - 2.03i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.88 + 3.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.0 + 6.93i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 - 3.80iT - 73T^{2} \)
79 \( 1 + (-4.87 + 8.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.4 - 7.75i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.2iT - 89T^{2} \)
97 \( 1 + (-12.3 - 7.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.26577138727399604751578564660, −9.519277051619987216090314085194, −8.558652028382715271006112284269, −7.901903831256694000059969067808, −6.62116708625791276730270804417, −5.90174719694469427530351743986, −5.03940537939084339309765386042, −3.66301505355676002893849609758, −2.19689050196397268683540189184, −0.904200320819367982594390277195, 2.31154187556411723384603921487, 3.31801915294275123930448365420, 4.42494164478248697314647713436, 4.96531982373886706059676447532, 6.46742797269586180343451635917, 7.59653700887393300485899840997, 8.609709470532339833215312485183, 9.099920037038637847243655745651, 10.03599130910298612109865556906, 11.02170702547262459921454956627

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