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Properties

Label 1-23e2-529.315-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.372 + 0.928i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.980 + 0.197i)2-s + (0.717 + 0.696i)3-s + (0.922 + 0.386i)4-s + (−0.426 + 0.904i)5-s + (0.566 + 0.824i)6-s + (−0.311 + 0.950i)7-s + (0.827 + 0.561i)8-s + (0.0310 + 0.999i)9-s + (−0.596 + 0.802i)10-s + (0.00620 − 0.999i)11-s + (0.392 + 0.919i)12-s + (0.481 − 0.876i)13-s + (−0.492 + 0.870i)14-s + (−0.935 + 0.352i)15-s + (0.700 + 0.713i)16-s + (−0.215 + 0.976i)17-s + ⋯
L(s)  = 1  + (0.980 + 0.197i)2-s + (0.717 + 0.696i)3-s + (0.922 + 0.386i)4-s + (−0.426 + 0.904i)5-s + (0.566 + 0.824i)6-s + (−0.311 + 0.950i)7-s + (0.827 + 0.561i)8-s + (0.0310 + 0.999i)9-s + (−0.596 + 0.802i)10-s + (0.00620 − 0.999i)11-s + (0.392 + 0.919i)12-s + (0.481 − 0.876i)13-s + (−0.492 + 0.870i)14-s + (−0.935 + 0.352i)15-s + (0.700 + 0.713i)16-s + (−0.215 + 0.976i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.372 + 0.928i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.372 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.661520489 + 2.456560776i\)
\(L(\frac12)\) \(\approx\) \(1.661520489 + 2.456560776i\)
\(L(1)\) \(\approx\) \(1.780601688 + 1.220442260i\)
\(L(1)\) \(\approx\) \(1.780601688 + 1.220442260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.980 + 0.197i)T \)
3 \( 1 + (0.717 + 0.696i)T \)
5 \( 1 + (-0.426 + 0.904i)T \)
7 \( 1 + (-0.311 + 0.950i)T \)
11 \( 1 + (0.00620 - 0.999i)T \)
13 \( 1 + (0.481 - 0.876i)T \)
17 \( 1 + (-0.215 + 0.976i)T \)
19 \( 1 + (0.0558 - 0.998i)T \)
29 \( 1 + (0.988 + 0.148i)T \)
31 \( 1 + (-0.0186 - 0.999i)T \)
37 \( 1 + (-0.873 + 0.487i)T \)
41 \( 1 + (0.105 + 0.994i)T \)
43 \( 1 + (-0.616 - 0.787i)T \)
47 \( 1 + (-0.917 + 0.398i)T \)
53 \( 1 + (-0.596 - 0.802i)T \)
59 \( 1 + (0.227 - 0.973i)T \)
61 \( 1 + (0.645 + 0.763i)T \)
67 \( 1 + (0.275 + 0.961i)T \)
71 \( 1 + (0.626 - 0.779i)T \)
73 \( 1 + (0.988 - 0.148i)T \)
79 \( 1 + (0.867 - 0.498i)T \)
83 \( 1 + (-0.834 - 0.551i)T \)
89 \( 1 + (0.503 + 0.863i)T \)
97 \( 1 + (-0.999 + 0.0372i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.203838455040585234700859241538, −22.85124086351107572864949205594, −21.1324749273609068525112036844, −20.75391248664347687156603145260, −19.8991603860558442306361515816, −19.53384035823756350286162723737, −18.38257384896517266360287001650, −17.147128784818756972251022690218, −16.17097116253912900087240027679, −15.52559993134902558425265261260, −14.219588956974318281732893539104, −13.885331084363402622435229727650, −12.877152853144917244304730487747, −12.312430100745303026590742310761, −11.52613935853681137085949206334, −10.17172025517763497885941614647, −9.24509731506014907768934925043, −8.03004748832964351118326101587, −7.142249449491632956821003385661, −6.51978149605816169474467812500, −5.013357875593485867069784653105, −4.13057902218653279547063404505, −3.391157732092991760878336514870, −1.98622887156822905105758804225, −1.11536068959369810604886405436, 2.22428170236965383110727844363, 3.13796088408233038470989215474, 3.54782454827644520668627703694, 4.83368433883531151635296092781, 5.8782945562071817550545588354, 6.68493823148523747360194821199, 8.07172364863833354148056938992, 8.52302544568618241245388585949, 10.01506874452008781097952902649, 10.93825134377661202905780013172, 11.54013041445190230972372748849, 12.84391567256380418004393573222, 13.58961557975372377355924558453, 14.50706376470314955519669861895, 15.335555535206754700030721707534, 15.57560700287086332312477756675, 16.497112912973948218623647801, 17.84458117461854175155986149261, 19.127511093056549595183390727658, 19.5288942008343954518279694398, 20.59298924275762034248765139243, 21.57237143110216903950724518606, 22.0080794976012713614513434280, 22.60437720640198106004938805993, 23.70754547280112116746477533064

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