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Properties

Label 2-522-29.24-c1-0-1
Degree $2$
Conductor $522$
Sign $0.434 - 0.900i$
Analytic cond. $4.16819$
Root an. cond. $2.04161$
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  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (2.02 + 2.53i)5-s + (0.524 + 2.29i)7-s + (0.900 − 0.433i)8-s + (0.722 − 3.16i)10-s + (−3.42 − 1.64i)11-s + (3.69 + 1.77i)13-s + (1.46 − 1.84i)14-s + (−0.900 − 0.433i)16-s − 7.38·17-s + (−1.85 + 8.10i)19-s + (−2.92 + 1.40i)20-s + (0.846 + 3.70i)22-s + (1.86 − 2.33i)23-s + ⋯
L(s)  = 1  + (−0.440 − 0.552i)2-s + (−0.111 + 0.487i)4-s + (0.905 + 1.13i)5-s + (0.198 + 0.868i)7-s + (0.318 − 0.153i)8-s + (0.228 − 1.00i)10-s + (−1.03 − 0.497i)11-s + (1.02 + 0.493i)13-s + (0.392 − 0.492i)14-s + (−0.225 − 0.108i)16-s − 1.79·17-s + (−0.424 + 1.86i)19-s + (−0.654 + 0.315i)20-s + (0.180 + 0.790i)22-s + (0.388 − 0.487i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ 0.434 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978354 + 0.614446i\)
\(L(\frac12)\) \(\approx\) \(0.978354 + 0.614446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
3 \( 1 \)
29 \( 1 + (1.65 - 5.12i)T \)
good5 \( 1 + (-2.02 - 2.53i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-0.524 - 2.29i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (3.42 + 1.64i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-3.69 - 1.77i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 7.38T + 17T^{2} \)
19 \( 1 + (1.85 - 8.10i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (-1.86 + 2.33i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (3.92 + 4.92i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (-10.0 + 4.86i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 - 9.70T + 41T^{2} \)
43 \( 1 + (1 - 1.25i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-0.554 - 0.267i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-2.10 - 2.64i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 4.59T + 59T^{2} \)
61 \( 1 + (-1.51 - 6.64i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (10.2 - 4.93i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-1.44 - 0.695i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (2.21 - 2.77i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (-11.7 + 5.63i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (-1.55 + 6.83i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (2.35 + 2.95i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-2.00 + 8.78i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.87035581650972179645764841528, −10.39956524278262650918567526391, −9.236154212223788538839929532165, −8.626701151401999720763838520405, −7.53648280647009942511641288700, −6.27286077784589386146254374743, −5.74880767178454121094242387739, −4.06782714312450032154975345324, −2.69739902363766771889467831457, −2.00328907623305157052326636524, 0.78364208134152709725964072594, 2.28374952310875488764899959020, 4.40013151265473375230316712563, 5.06044902362176423991197167543, 6.13871771247122304570921565336, 7.11462949970411685418884061269, 8.123672182414108392520425591859, 8.974903068954495603718130953635, 9.551618823478830301929180154257, 10.72116244201535316733115218199

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