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Properties

Label 2-539-7.2-c1-0-29
Degree $2$
Conductor $539$
Sign $-0.968 + 0.250i$
Analytic cond. $4.30393$
Root an. cond. $2.07459$
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 − 1.73i)2-s + (−0.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (0.5 − 0.866i)5-s − 1.99·6-s + (1 − 1.73i)9-s + (−0.999 − 1.73i)10-s + (−0.5 − 0.866i)11-s + (−1 + 1.73i)12-s − 4·13-s − 0.999·15-s + (1.99 − 3.46i)16-s + (−1 − 1.73i)17-s + (−2 − 3.46i)18-s − 1.99·20-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.288 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (0.223 − 0.387i)5-s − 0.816·6-s + (0.333 − 0.577i)9-s + (−0.316 − 0.547i)10-s + (−0.150 − 0.261i)11-s + (−0.288 + 0.499i)12-s − 1.10·13-s − 0.258·15-s + (0.499 − 0.866i)16-s + (−0.242 − 0.420i)17-s + (−0.471 − 0.816i)18-s − 0.447·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.241450 - 1.89471i\)
\(L(\frac12)\) \(\approx\) \(0.241450 - 1.89471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−10.58260609971970333494181149261, −9.824064317429458425643783876155, −8.931063904650405592360426580297, −7.56645059217730979516684969241, −6.71699091693613544161400637970, −5.39455014640119999984840278926, −4.63122778170841813451460445300, −3.41957232484765275166904434092, −2.27302924596790538133867218119, −0.968236289550503584798729107264, 2.32601443659742790056971909499, 4.07236695748109229217440871245, 4.82097633270886272408152654355, 5.63578811875978607975840195731, 6.60237174557722222786844192066, 7.42275819568418729090796066271, 8.162741939147974914382345281614, 9.571415605014629107687188324938, 10.32467625614385717755264612114, 11.06721126792996225281890271758

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