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Properties

Label 2-539-539.461-c1-0-37
Degree $2$
Conductor $539$
Sign $0.659 - 0.751i$
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  + (−1.00 + 2.08i)2-s + (2.98 − 0.680i)3-s + (−2.08 − 2.61i)4-s + (−0.377 + 0.0860i)5-s + (−1.57 + 6.89i)6-s + (−1.63 − 2.08i)7-s + (3.04 − 0.695i)8-s + (5.72 − 2.75i)9-s + (0.199 − 0.872i)10-s + (1.70 − 2.84i)11-s + (−8.01 − 6.39i)12-s + (5.99 + 2.88i)13-s + (5.97 − 1.30i)14-s + (−1.06 + 0.513i)15-s + (−0.117 + 0.512i)16-s + (2.03 − 2.55i)17-s + ⋯
L(s)  = 1  + (−0.709 + 1.47i)2-s + (1.72 − 0.392i)3-s + (−1.04 − 1.30i)4-s + (−0.168 + 0.0384i)5-s + (−0.642 + 2.81i)6-s + (−0.616 − 0.787i)7-s + (1.07 − 0.245i)8-s + (1.90 − 0.919i)9-s + (0.0629 − 0.275i)10-s + (0.515 − 0.857i)11-s + (−2.31 − 1.84i)12-s + (1.66 + 0.800i)13-s + (1.59 − 0.349i)14-s + (−0.275 + 0.132i)15-s + (−0.0292 + 0.128i)16-s + (0.493 − 0.618i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.659 - 0.751i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52215 + 0.689484i\)
\(L(\frac12)\) \(\approx\) \(1.52215 + 0.689484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.63 + 2.08i)T \)
11 \( 1 + (-1.70 + 2.84i)T \)
good2 \( 1 + (1.00 - 2.08i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (-2.98 + 0.680i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (0.377 - 0.0860i)T + (4.50 - 2.16i)T^{2} \)
13 \( 1 + (-5.99 - 2.88i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-2.03 + 2.55i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + 2.79T + 19T^{2} \)
23 \( 1 + (-1.63 - 2.04i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-2.63 - 2.09i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 + 0.666iT - 31T^{2} \)
37 \( 1 + (4.21 - 5.28i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (0.975 + 4.27i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (2.09 + 0.478i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (-5.05 + 10.5i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (-5.65 - 7.09i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-3.16 - 0.721i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (3.44 - 4.31i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + 3.83T + 67T^{2} \)
71 \( 1 + (6.42 + 8.05i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (4.48 - 2.15i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 - 8.61iT - 79T^{2} \)
83 \( 1 + (11.3 - 5.45i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (6.06 + 12.5i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−10.44320428768614100408248295458, −9.473658739926343900778142784852, −8.758414713811803125595726119536, −8.415787664374749290815022339423, −7.33235348189245258211510101494, −6.85698385277276905482699756193, −5.92511965634415002409589036175, −4.03393997546760324462344036850, −3.25883457658192417528222215162, −1.23569701206824311399981240713, 1.62226840147747260614952177001, 2.65534412752623379460294106275, 3.50972725635578351858097673702, 4.18683019410633735571669518842, 6.18174440040533039383435481904, 7.76988453998101469711268002914, 8.694907839821934144717066220175, 8.820388875850890885058408535705, 9.937227606900503231267679211787, 10.27429046450853894726280479060

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