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Properties

Label 2-539-11.3-c1-0-26
Degree $2$
Conductor $539$
Sign $0.152 + 0.988i$
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  + (1.38 − 1.00i)2-s + (−0.708 − 2.17i)3-s + (0.282 − 0.869i)4-s + (3.28 + 2.39i)5-s + (−3.16 − 2.29i)6-s + (0.572 + 1.76i)8-s + (−1.82 + 1.32i)9-s + 6.94·10-s + (−0.582 − 3.26i)11-s − 2.09·12-s + (2.65 − 1.92i)13-s + (2.87 − 8.86i)15-s + (4.03 + 2.93i)16-s + (−1.06 − 0.776i)17-s + (−1.18 + 3.65i)18-s + (−0.668 − 2.05i)19-s + ⋯
L(s)  = 1  + (0.976 − 0.709i)2-s + (−0.408 − 1.25i)3-s + (0.141 − 0.434i)4-s + (1.47 + 1.06i)5-s + (−1.29 − 0.938i)6-s + (0.202 + 0.623i)8-s + (−0.607 + 0.441i)9-s + 2.19·10-s + (−0.175 − 0.984i)11-s − 0.604·12-s + (0.735 − 0.534i)13-s + (0.743 − 2.28i)15-s + (1.00 + 0.733i)16-s + (−0.259 − 0.188i)17-s + (−0.279 + 0.861i)18-s + (−0.153 − 0.471i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.152 + 0.988i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.152 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99228 - 1.70924i\)
\(L(\frac12)\) \(\approx\) \(1.99228 - 1.70924i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.582 + 3.26i)T \)
good2 \( 1 + (-1.38 + 1.00i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.708 + 2.17i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-3.28 - 2.39i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-2.65 + 1.92i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.06 + 0.776i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.668 + 2.05i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.86T + 23T^{2} \)
29 \( 1 + (0.0754 - 0.232i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.55 - 4.03i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.0789 + 0.243i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.77 - 5.45i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + (-1.25 - 3.86i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.04 + 2.94i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.303 + 0.935i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.49 - 1.08i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.00T + 67T^{2} \)
71 \( 1 + (5.23 + 3.80i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.98 + 9.17i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.47 + 3.25i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.67 - 1.21i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + (2.09 - 1.51i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.97143413735094053813853075981, −10.19320035599303674879958179053, −8.843120838726406071891481558517, −7.70145061481480128926133853514, −6.58285566289710845570989775970, −6.00789076628410007847464219681, −5.25003147477634991357494723746, −3.44019566653625818595452838337, −2.55425292495873370962177805337, −1.54144732554376161397862655206, 1.79571673833905288170118289491, 3.96749051943242744946303487229, 4.57104430419637073028388578151, 5.46174727212718115051312063274, 5.87434315002659721432564460207, 6.98644386997808171271030599277, 8.566240947911115759369197740821, 9.568574052714503961226961378084, 9.911590299682718576970681377953, 10.78447002078206178689988422078

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