Properties

Label 1-23e2-529.25-r0-0-0
Degree $1$
Conductor $529$
Sign $0.159 - 0.987i$
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  + (0.251 − 0.967i)2-s + (0.922 + 0.386i)3-s + (−0.873 − 0.487i)4-s + (0.999 + 0.0248i)5-s + (0.606 − 0.794i)6-s + (−0.998 − 0.0620i)7-s + (−0.691 + 0.722i)8-s + (0.700 + 0.713i)9-s + (0.275 − 0.961i)10-s + (0.154 − 0.987i)11-s + (−0.616 − 0.787i)12-s + (0.00620 − 0.999i)13-s + (−0.311 + 0.950i)14-s + (0.912 + 0.409i)15-s + (0.524 + 0.851i)16-s + (0.751 + 0.659i)17-s + ⋯
L(s)  = 1  + (0.251 − 0.967i)2-s + (0.922 + 0.386i)3-s + (−0.873 − 0.487i)4-s + (0.999 + 0.0248i)5-s + (0.606 − 0.794i)6-s + (−0.998 − 0.0620i)7-s + (−0.691 + 0.722i)8-s + (0.700 + 0.713i)9-s + (0.275 − 0.961i)10-s + (0.154 − 0.987i)11-s + (−0.616 − 0.787i)12-s + (0.00620 − 0.999i)13-s + (−0.311 + 0.950i)14-s + (0.912 + 0.409i)15-s + (0.524 + 0.851i)16-s + (0.751 + 0.659i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.159 - 0.987i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.159 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.639409348 - 1.395182281i\)
\(L(\frac12)\) \(\approx\) \(1.639409348 - 1.395182281i\)
\(L(1)\) \(\approx\) \(1.433789191 - 0.7358406390i\)
\(L(1)\) \(\approx\) \(1.433789191 - 0.7358406390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.251 - 0.967i)T \)
3 \( 1 + (0.922 + 0.386i)T \)
5 \( 1 + (0.999 + 0.0248i)T \)
7 \( 1 + (-0.998 - 0.0620i)T \)
11 \( 1 + (0.154 - 0.987i)T \)
13 \( 1 + (0.00620 - 0.999i)T \)
17 \( 1 + (0.751 + 0.659i)T \)
19 \( 1 + (0.984 - 0.172i)T \)
29 \( 1 + (-0.834 - 0.551i)T \)
31 \( 1 + (-0.449 - 0.893i)T \)
37 \( 1 + (-0.986 + 0.160i)T \)
41 \( 1 + (0.481 - 0.876i)T \)
43 \( 1 + (0.783 + 0.621i)T \)
47 \( 1 + (0.682 - 0.730i)T \)
53 \( 1 + (0.275 + 0.961i)T \)
59 \( 1 + (-0.514 - 0.857i)T \)
61 \( 1 + (-0.966 + 0.257i)T \)
67 \( 1 + (0.645 + 0.763i)T \)
71 \( 1 + (-0.935 + 0.352i)T \)
73 \( 1 + (-0.834 + 0.551i)T \)
79 \( 1 + (0.890 - 0.454i)T \)
83 \( 1 + (0.437 - 0.899i)T \)
89 \( 1 + (0.586 + 0.809i)T \)
97 \( 1 + (-0.596 + 0.802i)T \)
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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

−23.872957053975021832090511242745, −22.87577246676974089211420140725, −22.15915654268828623872973900996, −21.19954435392537870900076012235, −20.447452485638362003670670600006, −19.298058797617591264867877144135, −18.43052043882597477351901512501, −17.86093084836606349302224314927, −16.70919223772237247287720877393, −16.050677139584813197671335611694, −15.00366921502928413918449553216, −14.110539870416964890421525238869, −13.76054554013523790905137592113, −12.70687123456670637977326998268, −12.20742821784733217481640754828, −10.08039185572345678203718508076, −9.330992281429528750755291809824, −9.02528052628836878079317418888, −7.45213128085954287127219212117, −6.99527128570316649796950169201, −6.068685500252509683937674975387, −4.98389312337412789679904461576, −3.73052039750177124809859177470, −2.79849206501904454868198954357, −1.48109250541521228799965977109, 1.085437385075732515226810210843, 2.40179349035268795220590645493, 3.19174399817613866848469335386, 3.84736355640298052053015214313, 5.39615658053455867710457563931, 5.97375508355871964105629562560, 7.6319197201326477387408889633, 8.83871201196185406012241986884, 9.45916067451738198764086032160, 10.20734960978865669323392493564, 10.83699086801369239655584580077, 12.32243300749258711917823317250, 13.207553021321059176217388188736, 13.63832942111847697516182040702, 14.46295323819350247773294131844, 15.42082776595602593738135353223, 16.5179815605855103931462685088, 17.51097047130497008006847445116, 18.76058055551771889013055448094, 19.07096048836386321748952976437, 20.20050011125902792066414403250, 20.64935821193521172358832410890, 21.66096531996394413659456342922, 22.10914391992432209496074617824, 22.879065114323095621449572429945

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