Properties

Label 2-522-29.25-c1-0-9
Degree $2$
Conductor $522$
Sign $-0.451 + 0.892i$
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  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (0.788 − 3.45i)5-s + (3.72 − 1.79i)7-s + (−0.623 + 0.781i)8-s + (−3.19 − 1.53i)10-s + (1.14 + 1.43i)11-s + (2.09 + 2.62i)13-s + (−0.920 − 4.03i)14-s + (0.623 + 0.781i)16-s − 3.52·17-s + (−2.45 − 1.18i)19-s + (−2.20 + 2.77i)20-s + (1.65 − 0.799i)22-s + (−0.679 − 2.97i)23-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (−0.450 − 0.216i)4-s + (0.352 − 1.54i)5-s + (1.40 − 0.678i)7-s + (−0.220 + 0.276i)8-s + (−1.00 − 0.486i)10-s + (0.346 + 0.434i)11-s + (0.580 + 0.728i)13-s + (−0.246 − 1.07i)14-s + (0.155 + 0.195i)16-s − 0.853·17-s + (−0.563 − 0.271i)19-s + (−0.494 + 0.619i)20-s + (0.353 − 0.170i)22-s + (−0.141 − 0.620i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(522\)    =    \(2 \cdot 3^{2} \cdot 29\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(4.16819\)
Root analytic conductor: \(2.04161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{522} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 522,\ (\ :1/2),\ -0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928061 - 1.50993i\)
\(L(\frac12)\) \(\approx\) \(0.928061 - 1.50993i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 \)
29 \( 1 + (0.127 - 5.38i)T \)
good5 \( 1 + (-0.788 + 3.45i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-3.72 + 1.79i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-1.14 - 1.43i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.09 - 2.62i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 3.52T + 17T^{2} \)
19 \( 1 + (2.45 + 1.18i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (0.679 + 2.97i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (0.196 - 0.861i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-3.04 + 3.82i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 - 3.01T + 41T^{2} \)
43 \( 1 + (0.409 + 1.79i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (1.25 + 1.57i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (1.47 - 6.46i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 6.12T + 59T^{2} \)
61 \( 1 + (1.64 - 0.792i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (0.0862 - 0.108i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-8.17 - 10.2i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-3.42 - 15.0i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (9.90 - 12.4i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (0.0422 + 0.0203i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.800 - 3.50i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (4.71 + 2.27i)T + (60.4 + 75.8i)T^{2} \)
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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

−10.84775713313874785467292825314, −9.639868958487797651161601778202, −8.779795903353926648398038934700, −8.328719207405277140444706613512, −6.96134159463029807639786697345, −5.55678518493014800128621329207, −4.47829572489867323970317579375, −4.27892057608169473788891171706, −2.01175577368104247586682742279, −1.14389184461479914509949922324, 2.07402907609481607370222249293, 3.34835307621649330921868472230, 4.66555860366483474385345315852, 5.91384070987123135130806409199, 6.36274059617314804174226567738, 7.61365695883592020161039942586, 8.230344917357805374715762991929, 9.236288142583999561707914917956, 10.41819397376853441607235096947, 11.16269746336017376434033030520

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