Properties

Label 1-23e2-529.495-r0-0-0
Degree $1$
Conductor $529$
Sign $0.971 - 0.236i$
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  + (−0.996 − 0.0868i)2-s + (0.0558 + 0.998i)3-s + (0.984 + 0.172i)4-s + (0.955 − 0.293i)5-s + (0.0310 − 0.999i)6-s + (0.735 − 0.678i)7-s + (−0.966 − 0.257i)8-s + (−0.993 + 0.111i)9-s + (−0.977 + 0.209i)10-s + (−0.287 − 0.957i)11-s + (−0.117 + 0.993i)12-s + (0.997 − 0.0744i)13-s + (−0.791 + 0.611i)14-s + (0.346 + 0.938i)15-s + (0.940 + 0.340i)16-s + (−0.709 − 0.704i)17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0868i)2-s + (0.0558 + 0.998i)3-s + (0.984 + 0.172i)4-s + (0.955 − 0.293i)5-s + (0.0310 − 0.999i)6-s + (0.735 − 0.678i)7-s + (−0.966 − 0.257i)8-s + (−0.993 + 0.111i)9-s + (−0.977 + 0.209i)10-s + (−0.287 − 0.957i)11-s + (−0.117 + 0.993i)12-s + (0.997 − 0.0744i)13-s + (−0.791 + 0.611i)14-s + (0.346 + 0.938i)15-s + (0.940 + 0.340i)16-s + (−0.709 − 0.704i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.971 - 0.236i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (495, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.971 - 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.102822429 - 0.1322108577i\)
\(L(\frac12)\) \(\approx\) \(1.102822429 - 0.1322108577i\)
\(L(1)\) \(\approx\) \(0.8953581153 + 0.02808820189i\)
\(L(1)\) \(\approx\) \(0.8953581153 + 0.02808820189i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.996 - 0.0868i)T \)
3 \( 1 + (0.0558 + 0.998i)T \)
5 \( 1 + (0.955 - 0.293i)T \)
7 \( 1 + (0.735 - 0.678i)T \)
11 \( 1 + (-0.287 - 0.957i)T \)
13 \( 1 + (0.997 - 0.0744i)T \)
17 \( 1 + (-0.709 - 0.704i)T \)
19 \( 1 + (-0.492 + 0.870i)T \)
29 \( 1 + (0.751 - 0.659i)T \)
31 \( 1 + (0.767 - 0.640i)T \)
37 \( 1 + (-0.358 + 0.933i)T \)
41 \( 1 + (0.969 + 0.245i)T \)
43 \( 1 + (-0.191 - 0.981i)T \)
47 \( 1 + (-0.917 - 0.398i)T \)
53 \( 1 + (-0.977 - 0.209i)T \)
59 \( 1 + (0.980 + 0.197i)T \)
61 \( 1 + (-0.999 + 0.0124i)T \)
67 \( 1 + (-0.535 + 0.844i)T \)
71 \( 1 + (-0.381 - 0.924i)T \)
73 \( 1 + (0.751 + 0.659i)T \)
79 \( 1 + (0.813 - 0.581i)T \)
83 \( 1 + (0.664 + 0.747i)T \)
89 \( 1 + (0.323 + 0.946i)T \)
97 \( 1 + (0.179 - 0.983i)T \)
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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

−23.848772562769106388988387965303, −22.86087246598267640279476638938, −21.51209635490034552912692921161, −20.94593231232390960575006943962, −19.92080014630590990016772463845, −19.12060997323384120224513416431, −18.13705881507186993235693193241, −17.78487141553527796392561907898, −17.35972563355179638582231245241, −15.94207117894870982983741209120, −14.98148793234675064984304693577, −14.23777455569561925267225310533, −13.07894257411235863336975272040, −12.31198341764874092339524373316, −11.16481448806968065042703173291, −10.61069746792627432004125456632, −9.24594785706796970260191975997, −8.65194253065164305813881710940, −7.756763194945596572111539209651, −6.63442919087379723925905395372, −6.17865647136730097058740836680, −4.99857575800546383244628091903, −2.84251056284374966994422062471, −2.03845898489351690239630381888, −1.367932491728956811088555263792, 0.87368918132646075379818091530, 2.16684209048644301318427588925, 3.316918591713610920624225719234, 4.54080102325272847967846345971, 5.72528454719139561950552208627, 6.49488958483923992730540347935, 8.133432463964295868156167457090, 8.54482867860510193587669512536, 9.56100258589045198206384787056, 10.38127165368403390164112326786, 10.94872945785706765696503953651, 11.77418580654602078303004742197, 13.40217123208541982302650227655, 14.04531961212713567057663193098, 15.171286477706693960142040561881, 16.13093362796363592275293889171, 16.702744724064343293824803185717, 17.48470863265656027820066261317, 18.17609832241395140324493161353, 19.249099704176958970180548354048, 20.394867810028036551877013814, 20.915192983370853805013371339, 21.26090525557755137070707750725, 22.379480253174157528642099260804, 23.58157128437294899176624663491

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