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Properties

Label 2-539-11.9-c1-0-18
Degree $2$
Conductor $539$
Sign $0.654 + 0.756i$
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  + (0.549 − 1.69i)2-s + (0.5 − 0.363i)3-s + (−0.938 − 0.681i)4-s + (0.858 + 2.64i)5-s + (−0.339 − 1.04i)6-s + (1.20 − 0.878i)8-s + (−0.809 + 2.48i)9-s + 4.93·10-s + (1.14 + 3.11i)11-s − 0.716·12-s + (1.32 − 4.08i)13-s + (1.38 + 1.00i)15-s + (−1.53 − 4.73i)16-s + (0.851 + 2.62i)17-s + (3.76 + 2.73i)18-s + (1.56 − 1.13i)19-s + ⋯
L(s)  = 1  + (0.388 − 1.19i)2-s + (0.288 − 0.209i)3-s + (−0.469 − 0.340i)4-s + (0.383 + 1.18i)5-s + (−0.138 − 0.426i)6-s + (0.427 − 0.310i)8-s + (−0.269 + 0.829i)9-s + 1.56·10-s + (0.346 + 0.938i)11-s − 0.206·12-s + (0.367 − 1.13i)13-s + (0.358 + 0.260i)15-s + (−0.384 − 1.18i)16-s + (0.206 + 0.635i)17-s + (0.887 + 0.644i)18-s + (0.359 − 0.261i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.654 + 0.756i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.654 + 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02644 - 0.926026i\)
\(L(\frac12)\) \(\approx\) \(2.02644 - 0.926026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 - 3.11i)T \)
good2 \( 1 + (-0.549 + 1.69i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.5 + 0.363i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.858 - 2.64i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.32 + 4.08i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.851 - 2.62i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.56 + 1.13i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4.37T + 23T^{2} \)
29 \( 1 + (6.98 + 5.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.0619 + 0.190i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.837 + 0.608i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.77 + 5.64i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + (10.5 - 7.66i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.20 - 3.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.92 + 5.02i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.305 - 0.940i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.41T + 67T^{2} \)
71 \( 1 + (0.623 + 1.91i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.06 + 5.86i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.94 + 5.98i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.531 - 1.63i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (3.58 - 11.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.81096357779093561915580538493, −10.19766239875816713393524858299, −9.310782673238675626226952547361, −7.82921378919245524191505151972, −7.23722601376690724192629925915, −6.03266346492605263044933383065, −4.79305102091204154657700947971, −3.46935005156872991064665792146, −2.69985767958160594318356433119, −1.73441131439206732589481871295, 1.38823752775102821621737059872, 3.44270425144872427156905038741, 4.61023747686271912805371474434, 5.46778281891806963961607975782, 6.27582312548541448971251894845, 7.15371453647316542350934824504, 8.390889609038473504351154898408, 8.961225835613996232117054110555, 9.574886537076176383943854133766, 11.10985744845962915861436573855

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