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Properties

Label 2-539-49.44-c1-0-45
Degree $2$
Conductor $539$
Sign $-0.935 - 0.352i$
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  + (1.68 − 0.253i)2-s + (−2.67 − 1.82i)3-s + (0.863 − 0.266i)4-s + (0.261 − 3.49i)5-s + (−4.97 − 2.39i)6-s + (−0.917 − 2.48i)7-s + (−1.68 + 0.810i)8-s + (2.74 + 6.98i)9-s + (−0.446 − 5.95i)10-s + (0.365 − 0.930i)11-s + (−2.79 − 0.863i)12-s + (2.31 + 2.90i)13-s + (−2.17 − 3.94i)14-s + (−7.08 + 8.87i)15-s + (−4.12 + 2.81i)16-s + (−3.86 + 3.58i)17-s + ⋯
L(s)  = 1  + (1.19 − 0.179i)2-s + (−1.54 − 1.05i)3-s + (0.431 − 0.133i)4-s + (0.117 − 1.56i)5-s + (−2.03 − 0.978i)6-s + (−0.346 − 0.938i)7-s + (−0.594 + 0.286i)8-s + (0.913 + 2.32i)9-s + (−0.141 − 1.88i)10-s + (0.110 − 0.280i)11-s + (−0.808 − 0.249i)12-s + (0.641 + 0.804i)13-s + (−0.581 − 1.05i)14-s + (−1.82 + 2.29i)15-s + (−1.03 + 0.702i)16-s + (−0.936 + 0.868i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.935 - 0.352i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.935 - 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.173660 + 0.952700i\)
\(L(\frac12)\) \(\approx\) \(0.173660 + 0.952700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.917 + 2.48i)T \)
11 \( 1 + (-0.365 + 0.930i)T \)
good2 \( 1 + (-1.68 + 0.253i)T + (1.91 - 0.589i)T^{2} \)
3 \( 1 + (2.67 + 1.82i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (-0.261 + 3.49i)T + (-4.94 - 0.745i)T^{2} \)
13 \( 1 + (-2.31 - 2.90i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (3.86 - 3.58i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (-2.27 + 3.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.40 - 2.22i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (1.53 + 6.74i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (4.61 + 7.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.63 - 0.503i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (2.59 - 1.24i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (5.17 + 2.49i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-3.50 + 0.528i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (-3.76 + 1.16i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-0.439 - 5.85i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (5.60 + 1.72i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (-2.56 - 4.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.504 + 2.20i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-1.48 - 0.224i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-4.00 + 6.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.35 + 5.46i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (4.87 + 12.4i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−10.97618717961473015976468831776, −9.478311927759358946976442639009, −8.448349233853051699137829936179, −7.21801981720423580525874223858, −6.25294223514070045212853366631, −5.64571653846296682023677408446, −4.67029349453610694765471609455, −4.07361602589377864366741812165, −1.77475101427364669847613667615, −0.46294279130362477912436164034, 3.02268246928183365171009093139, 3.71263219798980843575705773087, 5.05928411954655844435235525818, 5.55988882853973633394802032829, 6.47603341268150062656959196839, 6.89571083405657414969224303351, 9.026386655781500162964705234520, 9.859925384382574178503997981927, 10.74063748301585530642609078500, 11.24388415844039582300312085558

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