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Properties

Label 1-23e2-529.192-r0-0-0
Degree $1$
Conductor $529$
Sign $0.737 - 0.674i$
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  + (−0.673 − 0.739i)2-s + (−0.952 + 0.305i)3-s + (−0.0929 + 0.995i)4-s + (−0.820 + 0.571i)5-s + (0.867 + 0.498i)6-s + (0.998 + 0.0496i)7-s + (0.798 − 0.601i)8-s + (0.813 − 0.581i)9-s + (0.975 + 0.221i)10-s + (0.992 + 0.123i)11-s + (−0.215 − 0.976i)12-s + (−0.806 − 0.591i)13-s + (−0.635 − 0.771i)14-s + (0.606 − 0.794i)15-s + (−0.982 − 0.185i)16-s + (−0.358 − 0.933i)17-s + ⋯
L(s)  = 1  + (−0.673 − 0.739i)2-s + (−0.952 + 0.305i)3-s + (−0.0929 + 0.995i)4-s + (−0.820 + 0.571i)5-s + (0.867 + 0.498i)6-s + (0.998 + 0.0496i)7-s + (0.798 − 0.601i)8-s + (0.813 − 0.581i)9-s + (0.975 + 0.221i)10-s + (0.992 + 0.123i)11-s + (−0.215 − 0.976i)12-s + (−0.806 − 0.591i)13-s + (−0.635 − 0.771i)14-s + (0.606 − 0.794i)15-s + (−0.982 − 0.185i)16-s + (−0.358 − 0.933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.737 - 0.674i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (192, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.737 - 0.674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5781080483 - 0.2244671593i\)
\(L(\frac12)\) \(\approx\) \(0.5781080483 - 0.2244671593i\)
\(L(1)\) \(\approx\) \(0.5710239366 - 0.1053371842i\)
\(L(1)\) \(\approx\) \(0.5710239366 - 0.1053371842i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.673 - 0.739i)T \)
3 \( 1 + (-0.952 + 0.305i)T \)
5 \( 1 + (-0.820 + 0.571i)T \)
7 \( 1 + (0.998 + 0.0496i)T \)
11 \( 1 + (0.992 + 0.123i)T \)
13 \( 1 + (-0.806 - 0.591i)T \)
17 \( 1 + (-0.358 - 0.933i)T \)
19 \( 1 + (0.437 + 0.899i)T \)
29 \( 1 + (-0.986 + 0.160i)T \)
31 \( 1 + (0.931 - 0.363i)T \)
37 \( 1 + (-0.726 + 0.687i)T \)
41 \( 1 + (-0.514 - 0.857i)T \)
43 \( 1 + (0.751 - 0.659i)T \)
47 \( 1 + (-0.334 - 0.942i)T \)
53 \( 1 + (0.975 - 0.221i)T \)
59 \( 1 + (-0.117 - 0.993i)T \)
61 \( 1 + (0.105 - 0.994i)T \)
67 \( 1 + (0.767 + 0.640i)T \)
71 \( 1 + (0.566 + 0.824i)T \)
73 \( 1 + (-0.986 - 0.160i)T \)
79 \( 1 + (-0.535 + 0.844i)T \)
83 \( 1 + (0.626 - 0.779i)T \)
89 \( 1 + (-0.426 + 0.904i)T \)
97 \( 1 + (0.735 - 0.678i)T \)
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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

−23.95668523302198352812508927182, −23.00990529922474515230476203888, −22.16725296076223192976010003018, −21.09752773710077585835105895722, −19.707178924603239140067334077434, −19.41720452015839370809726336839, −18.31080539543730318151230687050, −17.34541427099005170191241147858, −17.05507803931514062667260385644, −16.16722869810693583943231419067, −15.25731941061305717067898004109, −14.469712594010150429883273542897, −13.327790047047840906470106777, −12.05781743184507638017559596818, −11.45957117292163885228123701852, −10.71291158036868499689428330825, −9.41699778773080668148398693710, −8.55182414617452560936144707054, −7.58136207965367045522372092539, −6.94863417474031490546963662014, −5.86317872682887808813279147502, −4.79673043484580594203226814983, −4.28772243587829790704839509459, −1.82472783132635727562293935662, −0.92770411053603836279917847881, 0.66827063226748714105363567470, 2.00542120603827260777820820594, 3.45263277877026892861056199447, 4.29120444763494489801555736857, 5.267539616884377310849282994111, 6.86201318523514583449756705080, 7.483479322050205948656830520056, 8.52813620378245047179058299266, 9.72103919282827739185825956210, 10.45641448181728003369057238284, 11.44020330494294957499926206891, 11.76125215578245398481020164744, 12.47931713871209323939351302257, 14.00106752550457818910915490021, 15.042008203288908066666834761238, 15.88765685492171459566287937350, 16.97049688810278112899384155106, 17.46661182335777455526549274159, 18.39067797552525030403218072967, 18.96199235292126501543749893530, 20.176041221442869184253574467466, 20.668273279950319669376519026315, 21.86433909598212157551569960396, 22.42817289058935667137234056003, 22.984064755472726766960117792333

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