Properties

Label 2-522-9.4-c1-0-20
Degree $2$
Conductor $522$
Sign $-0.766 + 0.642i$
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  + (−0.5 − 0.866i)2-s + (1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−2 + 3.46i)5-s + (−1.5 − 0.866i)6-s + (−2 − 3.46i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s + 3.99·10-s + (−1.5 − 2.59i)11-s + 1.73i·12-s + (2 − 3.46i)13-s + (−1.99 + 3.46i)14-s + 6.92i·15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.894 + 1.54i)5-s + (−0.612 − 0.353i)6-s + (−0.755 − 1.30i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + 1.26·10-s + (−0.452 − 0.783i)11-s + 0.499i·12-s + (0.554 − 0.960i)13-s + (−0.534 + 0.925i)14-s + 1.78i·15-s + (−0.125 − 0.216i)16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.316233 - 0.868845i\)
\(L(\frac12)\) \(\approx\) \(0.316233 - 0.868845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (5.5 - 9.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.5 + 11.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.68814149194308779048142690642, −9.921832580486238912922497495286, −8.508348332336985268987266260528, −7.940753508420661579999946696512, −6.98980525457824654242203414080, −6.47939716990714207698124572533, −4.10033282277557981116912867359, −3.30892806257726009668805432868, −2.74522423752882798041695395374, −0.54778566110804784192606160938, 1.91579239802354757359501055724, 3.62636385839488074613739847691, 4.68445437112377909024392028044, 5.41901232190027349144128253144, 6.85879813496947167052722819407, 7.989957846964386755722021894134, 8.630231473453306395629624542908, 9.224159709218759284000746175305, 9.718935050625032138677210051807, 11.21680223536979672035758640741

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