Properties

Label 2-522-261.202-c1-0-22
Degree $2$
Conductor $522$
Sign $0.526 + 0.849i$
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  + (0.866 − 0.5i)2-s + (−0.866 + 1.5i)3-s + (0.499 − 0.866i)4-s + (1 − 1.73i)5-s + 1.73i·6-s + (−0.5 − 0.866i)7-s − 0.999i·8-s + (−1.5 − 2.59i)9-s − 1.99i·10-s + (0.866 − 0.5i)11-s + (0.866 + 1.49i)12-s + (2 − 3.46i)13-s + (−0.866 − 0.499i)14-s + (1.73 + 3i)15-s + (−0.5 − 0.866i)16-s − 2i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 + 0.866i)3-s + (0.249 − 0.433i)4-s + (0.447 − 0.774i)5-s + 0.707i·6-s + (−0.188 − 0.327i)7-s − 0.353i·8-s + (−0.5 − 0.866i)9-s − 0.632i·10-s + (0.261 − 0.150i)11-s + (0.250 + 0.433i)12-s + (0.554 − 0.960i)13-s + (−0.231 − 0.133i)14-s + (0.447 + 0.774i)15-s + (−0.125 − 0.216i)16-s − 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $0.526 + 0.849i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ 0.526 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55525 - 0.865620i\)
\(L(\frac12)\) \(\approx\) \(1.55525 - 0.865620i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 1.5i)T \)
29 \( 1 + (-3.33 + 4.23i)T \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (-3.46 - 2i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (1.73 + i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.79 - 4.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.2 - 6.5i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + (3.46 - 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 + (6.92 - 4i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.64922353592662416061022457918, −10.06113304460230431791905165455, −9.209570475432180066210776271631, −8.261435724526510588929701099229, −6.69273677992369047720059006923, −5.74726152148926872082731622915, −5.04312037863447606746042840224, −4.07191293077392805058901120471, −3.01070681632793610055022701291, −0.993592364786548688367774994223, 1.83182998458002102256063989657, 3.01540017284380410530336775221, 4.49826042255930150332052968818, 5.72975903247426715770396270385, 6.49185508220658472555346951209, 6.93538508916802539225330471461, 8.106003916460724126531369315752, 9.093596648162634530198979832565, 10.42095293830397189438433832842, 11.16521032756138033595203985268

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