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Properties

Label 2-522-29.12-c2-0-20
Degree $2$
Conductor $522$
Sign $-0.113 + 0.993i$
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 + 2.10i·5-s − 5.61·7-s + (−2 + 2i)8-s + (−2.10 + 2.10i)10-s + (−13.5 − 13.5i)11-s − 14.7i·13-s + (−5.61 − 5.61i)14-s − 4·16-s + (−3.75 − 3.75i)17-s + (−5.29 − 5.29i)19-s − 4.21·20-s − 27.0i·22-s − 2.47·23-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + 0.421i·5-s − 0.801·7-s + (−0.250 + 0.250i)8-s + (−0.210 + 0.210i)10-s + (−1.22 − 1.22i)11-s − 1.13i·13-s + (−0.400 − 0.400i)14-s − 0.250·16-s + (−0.221 − 0.221i)17-s + (−0.278 − 0.278i)19-s − 0.210·20-s − 1.22i·22-s − 0.107·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $-0.113 + 0.993i$
Analytic conductor: \(14.2234\)
Root analytic conductor: \(3.77140\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1),\ -0.113 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6919360853\)
\(L(\frac12)\) \(\approx\) \(0.6919360853\)
\(L(2)\) 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 \( 1 + (-13.4 - 25.6i)T \)
good5 \( 1 - 2.10iT - 25T^{2} \)
7 \( 1 + 5.61T + 49T^{2} \)
11 \( 1 + (13.5 + 13.5i)T + 121iT^{2} \)
13 \( 1 + 14.7iT - 169T^{2} \)
17 \( 1 + (3.75 + 3.75i)T + 289iT^{2} \)
19 \( 1 + (5.29 + 5.29i)T + 361iT^{2} \)
23 \( 1 + 2.47T + 529T^{2} \)
31 \( 1 + (21.1 + 21.1i)T + 961iT^{2} \)
37 \( 1 + (-31.1 + 31.1i)T - 1.36e3iT^{2} \)
41 \( 1 + (23.5 - 23.5i)T - 1.68e3iT^{2} \)
43 \( 1 + (15.4 + 15.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (12.0 - 12.0i)T - 2.20e3iT^{2} \)
53 \( 1 + 65.9T + 2.80e3T^{2} \)
59 \( 1 + 32.0T + 3.48e3T^{2} \)
61 \( 1 + (58.4 + 58.4i)T + 3.72e3iT^{2} \)
67 \( 1 - 20.4iT - 4.48e3T^{2} \)
71 \( 1 - 60.6iT - 5.04e3T^{2} \)
73 \( 1 + (-89.6 + 89.6i)T - 5.32e3iT^{2} \)
79 \( 1 + (80.1 + 80.1i)T + 6.24e3iT^{2} \)
83 \( 1 + 65.4T + 6.88e3T^{2} \)
89 \( 1 + (-118. - 118. i)T + 7.92e3iT^{2} \)
97 \( 1 + (108. - 108. i)T - 9.40e3iT^{2} \)
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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

−10.66257889467975583168668683676, −9.505299335275868591849106794488, −8.404642334617307207237668003429, −7.67523924341486758209028507079, −6.60422726697795398309018753154, −5.81682085913199202842722498465, −4.92778575928540955982962131531, −3.35360927058245817116412429132, −2.80584634501876190451771176095, −0.21516664725868451608225057176, 1.77753365956566112290668212683, 2.93589075134161069729937640464, 4.30052367924070688417131100415, 4.99493754682922848239182863755, 6.25153979723222082097531409796, 7.10013334211433572690496178815, 8.306976351699272694257849996653, 9.417954782794059037622809098523, 10.01893792129018429943681886111, 10.88156315422008841443374542395

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