Properties

Label 1-23e2-529.9-r0-0-0
Degree $1$
Conductor $529$
Sign $0.999 + 0.0118i$
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  + (−0.596 − 0.802i)2-s + (0.997 + 0.0744i)3-s + (−0.287 + 0.957i)4-s + (0.922 + 0.386i)5-s + (−0.535 − 0.844i)6-s + (0.545 + 0.837i)7-s + (0.940 − 0.340i)8-s + (0.988 + 0.148i)9-s + (−0.239 − 0.970i)10-s + (−0.791 + 0.611i)11-s + (−0.358 + 0.933i)12-s + (0.995 + 0.0991i)13-s + (0.346 − 0.938i)14-s + (0.890 + 0.454i)15-s + (−0.834 − 0.551i)16-s + (0.503 − 0.863i)17-s + ⋯
L(s)  = 1  + (−0.596 − 0.802i)2-s + (0.997 + 0.0744i)3-s + (−0.287 + 0.957i)4-s + (0.922 + 0.386i)5-s + (−0.535 − 0.844i)6-s + (0.545 + 0.837i)7-s + (0.940 − 0.340i)8-s + (0.988 + 0.148i)9-s + (−0.239 − 0.970i)10-s + (−0.791 + 0.611i)11-s + (−0.358 + 0.933i)12-s + (0.995 + 0.0991i)13-s + (0.346 − 0.938i)14-s + (0.890 + 0.454i)15-s + (−0.834 − 0.551i)16-s + (0.503 − 0.863i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.999 + 0.0118i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 + 0.0118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.785707248 + 0.01060497301i\)
\(L(\frac12)\) \(\approx\) \(1.785707248 + 0.01060497301i\)
\(L(1)\) \(\approx\) \(1.327594315 - 0.1215122067i\)
\(L(1)\) \(\approx\) \(1.327594315 - 0.1215122067i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.596 - 0.802i)T \)
3 \( 1 + (0.997 + 0.0744i)T \)
5 \( 1 + (0.922 + 0.386i)T \)
7 \( 1 + (0.545 + 0.837i)T \)
11 \( 1 + (-0.791 + 0.611i)T \)
13 \( 1 + (0.995 + 0.0991i)T \)
17 \( 1 + (0.503 - 0.863i)T \)
19 \( 1 + (-0.935 - 0.352i)T \)
29 \( 1 + (-0.996 + 0.0868i)T \)
31 \( 1 + (0.392 - 0.919i)T \)
37 \( 1 + (-0.847 - 0.530i)T \)
41 \( 1 + (-0.191 + 0.981i)T \)
43 \( 1 + (-0.263 - 0.964i)T \)
47 \( 1 + (0.854 - 0.519i)T \)
53 \( 1 + (-0.239 + 0.970i)T \)
59 \( 1 + (-0.709 - 0.704i)T \)
61 \( 1 + (-0.514 + 0.857i)T \)
67 \( 1 + (0.227 + 0.973i)T \)
71 \( 1 + (0.867 + 0.498i)T \)
73 \( 1 + (-0.996 - 0.0868i)T \)
79 \( 1 + (0.299 - 0.954i)T \)
83 \( 1 + (0.566 + 0.824i)T \)
89 \( 1 + (-0.820 + 0.571i)T \)
97 \( 1 + (-0.691 - 0.722i)T \)
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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

−23.954286915969959820990138671040, −22.99618312850963329457976473945, −21.417615346352334854011692246, −20.916233705536699883979656551218, −20.12945975002450351627375912474, −19.06326940002458595562856270466, −18.431806590959998405823745270883, −17.510381014703705845614074648332, −16.7586631261500014096047158826, −15.88656011509893806886678957270, −14.92280877510882613398018217365, −14.062471492029019058665944348371, −13.560641462397148310752246451216, −12.77787897787730004325294438644, −10.690989367981624511469376814340, −10.40578148909629399041023699322, −9.25815860145806554293774360524, −8.36190133632641396014657714018, −7.97914660323273316623134294428, −6.74059019242512802047424205139, −5.82092105084933954814656928991, −4.74033926003363192521920350824, −3.57096223685386659826000324217, −1.93988415832101888014258807340, −1.18773820274025605187954290923, 1.5890577549961153185798563926, 2.30243362077541325629360478571, 3.02057254708350500570788293735, 4.30411314801574808883495218532, 5.48361673102066671801676976077, 6.98007356303841740436228263956, 7.93994694204019080838067734783, 8.82487504552645113250014801035, 9.44939428968647975450655954935, 10.33479125699636188328331582199, 11.12710442960351830403806812163, 12.35110168445019805903987667603, 13.22641092235747961722000528755, 13.847119498790036956426871042580, 14.934543504065673883918466733953, 15.72585643842014246720881883771, 16.96110689879777266520151431786, 18.03621783454959078776581681854, 18.50899481253309335230294017582, 19.04528878376400866327584300730, 20.50213591088766664124802663367, 20.75773125096235010394492807940, 21.48117381004432599098898438509, 22.20405647661071458729372657315, 23.39979663880135403270963255553

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