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Properties

Label 2-539-49.46-c1-0-40
Degree $2$
Conductor $539$
Sign $-0.934 + 0.356i$
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  + (−0.0544 + 0.138i)2-s + (0.211 − 2.82i)3-s + (1.44 + 1.34i)4-s + (−2.51 − 1.71i)5-s + (0.379 + 0.182i)6-s + (−0.507 − 2.59i)7-s + (−0.533 + 0.257i)8-s + (−4.94 − 0.745i)9-s + (0.375 − 0.255i)10-s + (0.988 − 0.149i)11-s + (4.10 − 3.80i)12-s + (1.39 + 1.75i)13-s + (0.387 + 0.0708i)14-s + (−5.37 + 6.74i)15-s + (0.289 + 3.85i)16-s + (−2.68 − 0.828i)17-s + ⋯
L(s)  = 1  + (−0.0384 + 0.0980i)2-s + (0.122 − 1.62i)3-s + (0.724 + 0.672i)4-s + (−1.12 − 0.767i)5-s + (0.154 + 0.0746i)6-s + (−0.191 − 0.981i)7-s + (−0.188 + 0.0908i)8-s + (−1.64 − 0.248i)9-s + (0.118 − 0.0808i)10-s + (0.298 − 0.0449i)11-s + (1.18 − 1.09i)12-s + (0.387 + 0.485i)13-s + (0.103 + 0.0189i)14-s + (−1.38 + 1.74i)15-s + (0.0722 + 0.964i)16-s + (−0.651 − 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.934 + 0.356i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.193944 - 1.05091i\)
\(L(\frac12)\) \(\approx\) \(0.193944 - 1.05091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.507 + 2.59i)T \)
11 \( 1 + (-0.988 + 0.149i)T \)
good2 \( 1 + (0.0544 - 0.138i)T + (-1.46 - 1.36i)T^{2} \)
3 \( 1 + (-0.211 + 2.82i)T + (-2.96 - 0.447i)T^{2} \)
5 \( 1 + (2.51 + 1.71i)T + (1.82 + 4.65i)T^{2} \)
13 \( 1 + (-1.39 - 1.75i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (2.68 + 0.828i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (3.58 + 6.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.63 - 0.503i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (-1.03 - 4.53i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (0.113 - 0.197i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.64 + 4.31i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (4.78 - 2.30i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-8.95 - 4.31i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-3.04 + 7.76i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (9.91 + 9.20i)T + (3.96 + 52.8i)T^{2} \)
59 \( 1 + (2.71 - 1.85i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-6.33 + 5.87i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-7.24 + 12.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.11 + 13.6i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.54 - 14.1i)T + (-53.5 + 49.6i)T^{2} \)
79 \( 1 + (-1.42 - 2.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.62 + 4.54i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-1.76 - 0.266i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 9.39T + 97T^{2} \)
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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

−11.03875087977537255013634711505, −9.085540047737949922254288910628, −8.284678470207980755706051646061, −7.68445070806061615578438819122, −6.86161938926411197097839906680, −6.47594445461690180165565910163, −4.54439054043502752595828843780, −3.48565427784334863168470836842, −2.06056128742521428716160341087, −0.59349996378478848654035679745, 2.49152050479875121978519919434, 3.52000502759173392295721606759, 4.38037249086306297731519741836, 5.72100317244270060229720275416, 6.37658489962909426346003341259, 7.80576391860276058221316611979, 8.714186145730389468019588545866, 9.695747779735421422786565731744, 10.42674474535405657020344843169, 11.02985467499513850024071647477

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