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Properties

Label 2-539-77.37-c1-0-4
Degree $2$
Conductor $539$
Sign $-0.871 - 0.490i$
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  + (0.206 − 0.0920i)2-s + (0.214 − 0.0455i)3-s + (−1.30 + 1.44i)4-s + (0.260 + 2.48i)5-s + (0.0401 − 0.0291i)6-s + (−0.276 + 0.849i)8-s + (−2.69 + 1.20i)9-s + (0.282 + 0.488i)10-s + (3.04 + 1.32i)11-s + (−0.213 + 0.369i)12-s + (−4.15 − 3.01i)13-s + (0.168 + 0.520i)15-s + (−0.386 − 3.67i)16-s + (−1.31 − 0.584i)17-s + (−0.446 + 0.496i)18-s + (−4.06 − 4.50i)19-s + ⋯
L(s)  = 1  + (0.146 − 0.0650i)2-s + (0.123 − 0.0263i)3-s + (−0.652 + 0.724i)4-s + (0.116 + 1.10i)5-s + (0.0163 − 0.0118i)6-s + (−0.0975 + 0.300i)8-s + (−0.898 + 0.400i)9-s + (0.0892 + 0.154i)10-s + (0.917 + 0.398i)11-s + (−0.0616 + 0.106i)12-s + (−1.15 − 0.837i)13-s + (0.0436 + 0.134i)15-s + (−0.0965 − 0.918i)16-s + (−0.318 − 0.141i)17-s + (−0.105 + 0.116i)18-s + (−0.931 − 1.03i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.871 - 0.490i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.871 - 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212003 + 0.808201i\)
\(L(\frac12)\) \(\approx\) \(0.212003 + 0.808201i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.04 - 1.32i)T \)
good2 \( 1 + (-0.206 + 0.0920i)T + (1.33 - 1.48i)T^{2} \)
3 \( 1 + (-0.214 + 0.0455i)T + (2.74 - 1.22i)T^{2} \)
5 \( 1 + (-0.260 - 2.48i)T + (-4.89 + 1.03i)T^{2} \)
13 \( 1 + (4.15 + 3.01i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.31 + 0.584i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (4.06 + 4.50i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (3.54 - 6.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.01 - 6.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.803 - 7.64i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-3.89 - 0.828i)T + (33.8 + 15.0i)T^{2} \)
41 \( 1 + (-2.08 + 6.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.802T + 43T^{2} \)
47 \( 1 + (-4.51 - 5.01i)T + (-4.91 + 46.7i)T^{2} \)
53 \( 1 + (0.687 - 6.54i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (-1.92 + 2.13i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-0.0894 - 0.850i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (-0.823 - 1.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.65 - 2.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (9.93 - 11.0i)T + (-7.63 - 72.6i)T^{2} \)
79 \( 1 + (-2.24 + 0.997i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (1.81 - 1.32i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.867 - 1.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.77 - 7.09i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.19644358487807160596262623533, −10.38144282948582326945183837230, −9.320311465707708662901318551859, −8.597919363004570768451833778508, −7.50669125512018035016483588747, −6.87690438469904436228102203987, −5.54492696115765647845266403408, −4.50036101390426557977014235050, −3.23894264956812767575246690859, −2.49837240390187444206512914940, 0.44412191022686721696917398136, 2.08220107252664265161710746948, 4.07335795618048239258818143384, 4.60867399362784464429281249156, 5.88910316642342189232655706497, 6.38574644737845641632320408888, 8.117779447399976474493936563132, 8.817895080482535992187916731815, 9.426050068629196957283347647735, 10.19559139779561578261788408562

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