Properties

Label 2-522-9.4-c1-0-10
Degree $2$
Conductor $522$
Sign $0.790 - 0.612i$
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 + (0.837 + 1.51i)3-s + (−0.499 + 0.866i)4-s + (1.26 − 2.18i)5-s + (0.894 − 1.48i)6-s + (2.51 + 4.36i)7-s + 0.999·8-s + (−1.59 + 2.53i)9-s − 2.52·10-s + (−0.0574 − 0.0995i)11-s + (−1.73 − 0.0331i)12-s + (−1.56 + 2.70i)13-s + (2.51 − 4.36i)14-s + (4.37 + 0.0838i)15-s + (−0.5 − 0.866i)16-s − 1.03·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.483 + 0.875i)3-s + (−0.249 + 0.433i)4-s + (0.564 − 0.978i)5-s + (0.365 − 0.605i)6-s + (0.951 + 1.64i)7-s + 0.353·8-s + (−0.532 + 0.846i)9-s − 0.798·10-s + (−0.0173 − 0.0300i)11-s + (−0.499 − 0.00957i)12-s + (−0.433 + 0.750i)13-s + (0.673 − 1.16i)14-s + (1.12 + 0.0216i)15-s + (−0.125 − 0.216i)16-s − 0.251·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $0.790 - 0.612i$
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: \((2,\ 522,\ (\ :1/2),\ 0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47249 + 0.504208i\)
\(L(\frac12)\) \(\approx\) \(1.47249 + 0.504208i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.837 - 1.51i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.26 + 2.18i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.51 - 4.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.0574 + 0.0995i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.56 - 2.70i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.03T + 17T^{2} \)
19 \( 1 + 2.00T + 19T^{2} \)
23 \( 1 + (-1.32 + 2.28i)T + (-11.5 - 19.9i)T^{2} \)
31 \( 1 + (-4.30 + 7.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.08T + 37T^{2} \)
41 \( 1 + (3.92 - 6.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.02 - 5.23i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.522 - 0.904i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.62T + 53T^{2} \)
59 \( 1 + (-3.13 + 5.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.74 + 9.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.62 + 6.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 4.05T + 73T^{2} \)
79 \( 1 + (5.91 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.47 + 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.70T + 89T^{2} \)
97 \( 1 + (-1.74 - 3.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.05280653777908558572882531254, −9.698891633550531087391082581710, −9.311807648568954797951550560109, −8.555332735650828718761735443632, −8.020509442650103130306919632262, −6.04813185335392613372260249486, −4.96590008177938071446183846891, −4.46280974779879798155702996889, −2.68146536212990049865348485795, −1.87704627144603347241610808836, 1.06928982853189440070049213293, 2.51762201317205610363271376054, 3.98169394259650814036938485417, 5.39333131838790597780964258571, 6.66323171882748831024335435983, 7.13876292200398999333699385063, 7.85893590422130945783246697008, 8.698870505309875150521788122210, 10.05573671100081951681303511151, 10.52947839286596444773862737126

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