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Properties

Label 1-23e2-529.167-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.111 - 0.993i$
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  + (0.105 − 0.994i)2-s + (0.179 + 0.983i)3-s + (−0.977 − 0.209i)4-s + (−0.996 − 0.0868i)5-s + (0.997 − 0.0744i)6-s + (−0.215 + 0.976i)7-s + (−0.311 + 0.950i)8-s + (−0.935 + 0.352i)9-s + (−0.191 + 0.981i)10-s + (−0.239 − 0.970i)11-s + (0.0310 − 0.999i)12-s + (−0.691 − 0.722i)13-s + (0.948 + 0.317i)14-s + (−0.0929 − 0.995i)15-s + (0.912 + 0.409i)16-s + (0.813 − 0.581i)17-s + ⋯
L(s)  = 1  + (0.105 − 0.994i)2-s + (0.179 + 0.983i)3-s + (−0.977 − 0.209i)4-s + (−0.996 − 0.0868i)5-s + (0.997 − 0.0744i)6-s + (−0.215 + 0.976i)7-s + (−0.311 + 0.950i)8-s + (−0.935 + 0.352i)9-s + (−0.191 + 0.981i)10-s + (−0.239 − 0.970i)11-s + (0.0310 − 0.999i)12-s + (−0.691 − 0.722i)13-s + (0.948 + 0.317i)14-s + (−0.0929 − 0.995i)15-s + (0.912 + 0.409i)16-s + (0.813 − 0.581i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.111 - 0.993i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.111 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4605074932 - 0.5149383131i\)
\(L(\frac12)\) \(\approx\) \(0.4605074932 - 0.5149383131i\)
\(L(1)\) \(\approx\) \(0.7274963851 - 0.2090187509i\)
\(L(1)\) \(\approx\) \(0.7274963851 - 0.2090187509i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.105 - 0.994i)T \)
3 \( 1 + (0.179 + 0.983i)T \)
5 \( 1 + (-0.996 - 0.0868i)T \)
7 \( 1 + (-0.215 + 0.976i)T \)
11 \( 1 + (-0.239 - 0.970i)T \)
13 \( 1 + (-0.691 - 0.722i)T \)
17 \( 1 + (0.813 - 0.581i)T \)
19 \( 1 + (-0.820 + 0.571i)T \)
29 \( 1 + (0.890 + 0.454i)T \)
31 \( 1 + (0.664 - 0.747i)T \)
37 \( 1 + (-0.535 - 0.844i)T \)
41 \( 1 + (0.827 - 0.561i)T \)
43 \( 1 + (0.700 - 0.713i)T \)
47 \( 1 + (0.962 + 0.269i)T \)
53 \( 1 + (-0.191 - 0.981i)T \)
59 \( 1 + (-0.449 + 0.893i)T \)
61 \( 1 + (-0.791 - 0.611i)T \)
67 \( 1 + (0.995 - 0.0991i)T \)
71 \( 1 + (-0.952 - 0.305i)T \)
73 \( 1 + (0.890 - 0.454i)T \)
79 \( 1 + (0.0806 + 0.996i)T \)
83 \( 1 + (0.717 - 0.696i)T \)
89 \( 1 + (-0.986 - 0.160i)T \)
97 \( 1 + (-0.117 - 0.993i)T \)
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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

−23.49375676129613882428893880563, −23.39729099291139219125200883342, −22.57206240872385810904401210952, −21.256474429932469185337264861769, −20.00695610924125486351687641867, −19.34173518880322787694347498896, −18.73301287899700619393172237396, −17.48680194820117249410855943961, −17.12012662820655771181316635251, −16.07108565966193392531466375178, −15.06740792745366493708129498001, −14.39922770305430183868365249842, −13.575959434292094374139051000998, −12.54440070899531737893685604920, −12.11690994460564093778880916814, −10.66373836973822659538384057343, −9.52419780043096402777032445734, −8.33451464227825259171128174444, −7.665442811242135016556296220205, −7.00092695827909624399306326065, −6.37566717745687461475562990959, −4.79617406983092631074456501495, −4.07680012086097581570447826809, −2.82752459609182112761367000727, −1.06489651852342213479011925143, 0.42440418128934741403880619754, 2.52721563296747846618419190823, 3.1676461579906284015058213946, 4.06732081223292032436184610518, 5.118173222254707438966949046812, 5.81752417010330841891319118561, 7.87778787812566202830944295903, 8.56074701086177583265338195864, 9.35582625045147214263111373699, 10.38381812941651530179726054677, 11.051910969424530115061123414792, 12.057120823448109381134329730566, 12.52187155472396340307788110245, 13.92728456058296629576705868900, 14.73553580320960087018866401411, 15.52873548975162956320306757877, 16.31207597410025526002584018768, 17.3566012155463707306324650012, 18.65047000375568053969043014698, 19.21350189606150601668299167342, 19.89821560633686103347041104698, 20.94128551698684413741196001057, 21.36419458201220327637819021010, 22.42571830103212497257279811686, 22.779067949881766212604285270146

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