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Properties

Label 2-23e2-23.18-c1-0-10
Degree $2$
Conductor $529$
Sign $0.987 - 0.159i$
Analytic cond. $4.22408$
Root an. cond. $2.05525$
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  + (−0.226 − 0.0666i)2-s + (−0.313 + 0.361i)3-s + (−1.63 − 1.05i)4-s + (0.215 − 1.49i)5-s + (0.0952 − 0.0612i)6-s + (1.05 + 2.31i)7-s + (0.610 + 0.704i)8-s + (0.394 + 2.74i)9-s + (−0.148 + 0.325i)10-s + (−3.23 + 0.950i)11-s + (0.893 − 0.262i)12-s + (1.36 − 2.99i)13-s + (−0.0856 − 0.595i)14-s + (0.474 + 0.547i)15-s + (1.52 + 3.33i)16-s + (5.28 − 3.39i)17-s + ⋯
L(s)  = 1  + (−0.160 − 0.0471i)2-s + (−0.181 + 0.208i)3-s + (−0.817 − 0.525i)4-s + (0.0963 − 0.669i)5-s + (0.0388 − 0.0249i)6-s + (0.399 + 0.875i)7-s + (0.215 + 0.249i)8-s + (0.131 + 0.914i)9-s + (−0.0470 + 0.102i)10-s + (−0.975 + 0.286i)11-s + (0.257 − 0.0757i)12-s + (0.379 − 0.830i)13-s + (−0.0229 − 0.159i)14-s + (0.122 + 0.141i)15-s + (0.380 + 0.834i)16-s + (1.28 − 0.823i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(4.22408\)
Root analytic conductor: \(2.05525\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 529,\ (\ :1/2),\ 0.987 - 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09807 + 0.0881515i\)
\(L(\frac12)\) \(\approx\) \(1.09807 + 0.0881515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.226 + 0.0666i)T + (1.68 + 1.08i)T^{2} \)
3 \( 1 + (0.313 - 0.361i)T + (-0.426 - 2.96i)T^{2} \)
5 \( 1 + (-0.215 + 1.49i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-1.05 - 2.31i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (3.23 - 0.950i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-1.36 + 2.99i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-5.28 + 3.39i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-3.55 - 2.28i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-5.28 + 3.39i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-3.10 - 3.58i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-0.00540 - 0.0375i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.462 - 3.21i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-0.844 + 0.974i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 - 3.41T + 47T^{2} \)
53 \( 1 + (-2.79 - 6.11i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (4.32 - 9.47i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-2.35 - 2.72i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (6.54 + 1.92i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (3.92 + 1.15i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-3.66 - 2.35i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (0.997 - 2.18i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-0.420 - 2.92i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-8.09 + 9.34i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (1.51 - 10.5i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.49950422442695104095994716542, −10.12629460500436954219919991754, −9.114551061390286721930510281542, −8.243680524600343592487844411209, −7.66933533864219195389095767042, −5.66951376588728210849801540602, −5.31738753858142436632235334998, −4.59416968065632445457653163288, −2.80765319996144731604313275935, −1.16080511768565276636472720002, 0.952852496019363683051348340906, 3.10510044891143402238194107432, 3.97682174696524267305754796397, 5.13147473899212926698584901324, 6.40274557530091665463099120865, 7.32372872661060664380247753599, 8.041388758908750199497403601110, 9.037339293838806081154049425484, 10.02080130839516885056242331595, 10.67484453707000199502110356830

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