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Properties

Label 2-585-13.3-c1-0-5
Degree $2$
Conductor $585$
Sign $0.703 + 0.710i$
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  + (−1.06 − 1.83i)2-s + (−1.25 + 2.17i)4-s − 5-s + (−0.733 + 1.27i)7-s + 1.07·8-s + (1.06 + 1.83i)10-s + (0.172 + 0.298i)11-s + (1.76 + 3.14i)13-s + 3.11·14-s + (1.36 + 2.36i)16-s + (2.56 − 4.44i)17-s + (−0.253 + 0.438i)19-s + (1.25 − 2.17i)20-s + (0.365 − 0.632i)22-s + (0.537 + 0.930i)23-s + ⋯
L(s)  = 1  + (−0.750 − 1.29i)2-s + (−0.626 + 1.08i)4-s − 0.447·5-s + (−0.277 + 0.480i)7-s + 0.380·8-s + (0.335 + 0.581i)10-s + (0.0518 + 0.0898i)11-s + (0.490 + 0.871i)13-s + 0.832·14-s + (0.341 + 0.591i)16-s + (0.622 − 1.07i)17-s + (−0.0580 + 0.100i)19-s + (0.280 − 0.485i)20-s + (0.0778 − 0.134i)22-s + (0.112 + 0.194i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749059 - 0.312500i\)
\(L(\frac12)\) \(\approx\) \(0.749059 - 0.312500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
13 \( 1 + (-1.76 - 3.14i)T \)
good2 \( 1 + (1.06 + 1.83i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.733 - 1.27i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.172 - 0.298i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.56 + 4.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.253 - 0.438i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.537 - 0.930i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.02 - 6.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.17T + 31T^{2} \)
37 \( 1 + (1.36 + 2.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.06 - 5.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.93 + 5.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.26T + 47T^{2} \)
53 \( 1 - 4.07T + 53T^{2} \)
59 \( 1 + (2.53 - 4.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.58 + 4.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.598 + 1.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.222 + 0.386i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.52T + 73T^{2} \)
79 \( 1 - 0.834T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (-5.97 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.700 + 1.21i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60321896016353712209046775911, −9.767521689524044859069223751027, −9.037130095597541441943883650235, −8.375902329672482517400714404557, −7.20921483313004448559741001117, −6.09051467153307138622797421414, −4.68322111195474375075252477910, −3.45962494245081517311096089939, −2.55963128393530351015437525547, −1.10492623093241095047129026657, 0.75487914430656409040485524777, 3.12398047394544547838782229573, 4.38464944805874712355643041683, 5.74781761757980749438009697177, 6.39724126600876301270343782023, 7.37417001891157676813130459025, 8.170702306229724285185661143358, 8.610198755981969259115489825980, 9.920718376202744020713092293342, 10.36982084528965871752716593721

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