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Properties

Label 2-522-29.17-c2-0-11
Degree $2$
Conductor $522$
Sign $0.997 - 0.0646i$
Analytic cond. $14.2234$
Root an. cond. $3.77140$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s − 2i·4-s + 4.59i·5-s + 10.0·7-s + (−2 − 2i)8-s + (4.59 + 4.59i)10-s + (6.29 − 6.29i)11-s + 17.1i·13-s + (10.0 − 10.0i)14-s − 4·16-s + (−16.9 + 16.9i)17-s + (−2.81 + 2.81i)19-s + 9.18·20-s − 12.5i·22-s − 4.22·23-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s − 0.5i·4-s + 0.918i·5-s + 1.43·7-s + (−0.250 − 0.250i)8-s + (0.459 + 0.459i)10-s + (0.571 − 0.571i)11-s + 1.31i·13-s + (0.715 − 0.715i)14-s − 0.250·16-s + (−0.998 + 0.998i)17-s + (−0.148 + 0.148i)19-s + 0.459·20-s − 0.571i·22-s − 0.183·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $0.997 - 0.0646i$
Analytic conductor: \(14.2234\)
Root analytic conductor: \(3.77140\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1),\ 0.997 - 0.0646i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.695041014\)
\(L(\frac12)\) \(\approx\) \(2.695041014\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 \)
29 \( 1 + (-25.0 - 14.6i)T \)
good5 \( 1 - 4.59iT - 25T^{2} \)
7 \( 1 - 10.0T + 49T^{2} \)
11 \( 1 + (-6.29 + 6.29i)T - 121iT^{2} \)
13 \( 1 - 17.1iT - 169T^{2} \)
17 \( 1 + (16.9 - 16.9i)T - 289iT^{2} \)
19 \( 1 + (2.81 - 2.81i)T - 361iT^{2} \)
23 \( 1 + 4.22T + 529T^{2} \)
31 \( 1 + (-38.8 + 38.8i)T - 961iT^{2} \)
37 \( 1 + (-37.7 - 37.7i)T + 1.36e3iT^{2} \)
41 \( 1 + (-25.0 - 25.0i)T + 1.68e3iT^{2} \)
43 \( 1 + (-36.4 + 36.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (58.2 + 58.2i)T + 2.20e3iT^{2} \)
53 \( 1 + 21.1T + 2.80e3T^{2} \)
59 \( 1 - 46.7T + 3.48e3T^{2} \)
61 \( 1 + (27.3 - 27.3i)T - 3.72e3iT^{2} \)
67 \( 1 + 14.7iT - 4.48e3T^{2} \)
71 \( 1 + 5.13iT - 5.04e3T^{2} \)
73 \( 1 + (73.0 + 73.0i)T + 5.32e3iT^{2} \)
79 \( 1 + (-34.5 + 34.5i)T - 6.24e3iT^{2} \)
83 \( 1 + 111.T + 6.88e3T^{2} \)
89 \( 1 + (86.9 - 86.9i)T - 7.92e3iT^{2} \)
97 \( 1 + (-106. - 106. i)T + 9.40e3iT^{2} \)
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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

−11.00689647317489831168381112322, −10.06602648133888189461193851205, −8.872257664140957748268315230281, −8.078414084812253540592741792983, −6.75077141697734938474020423217, −6.14258652798912631542395063153, −4.67654264440979466836104003309, −4.02404126883164863924458464064, −2.56552814749011092301618782534, −1.49354037029967839873355841051, 1.05633857490629414151750504659, 2.65313604252836478669459418681, 4.51792778724977068038652386505, 4.71980427936513476264059648343, 5.83467786841141402834691641131, 7.06142329864766709268296581187, 8.025460894403037552278196574522, 8.587114675308733873325758483818, 9.584466309248916696700335481967, 10.86538654511279861451407763131

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