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Properties

Label 1-23e2-529.127-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: $\chi_{529} (127, \cdot )$
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

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

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