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Properties

Label 1-23e2-529.41-r0-0-0
Degree $1$
Conductor $529$
Sign $0.985 + 0.168i$
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  + (0.999 + 0.0248i)2-s + (−0.635 − 0.771i)3-s + (0.998 + 0.0496i)4-s + (−0.860 + 0.508i)5-s + (−0.616 − 0.787i)6-s + (−0.972 − 0.233i)7-s + (0.997 + 0.0744i)8-s + (−0.191 + 0.981i)9-s + (−0.873 + 0.487i)10-s + (−0.556 + 0.831i)11-s + (−0.596 − 0.802i)12-s + (0.606 − 0.794i)13-s + (−0.966 − 0.257i)14-s + (0.940 + 0.340i)15-s + (0.995 + 0.0991i)16-s + (0.975 + 0.221i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0248i)2-s + (−0.635 − 0.771i)3-s + (0.998 + 0.0496i)4-s + (−0.860 + 0.508i)5-s + (−0.616 − 0.787i)6-s + (−0.972 − 0.233i)7-s + (0.997 + 0.0744i)8-s + (−0.191 + 0.981i)9-s + (−0.873 + 0.487i)10-s + (−0.556 + 0.831i)11-s + (−0.596 − 0.802i)12-s + (0.606 − 0.794i)13-s + (−0.966 − 0.257i)14-s + (0.940 + 0.340i)15-s + (0.995 + 0.0991i)16-s + (0.975 + 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.985 + 0.168i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.985 + 0.168i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.658040942 + 0.1408763990i\)
\(L(\frac12)\) \(\approx\) \(1.658040942 + 0.1408763990i\)
\(L(1)\) \(\approx\) \(1.346741525 - 0.05132386166i\)
\(L(1)\) \(\approx\) \(1.346741525 - 0.05132386166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.999 + 0.0248i)T \)
3 \( 1 + (-0.635 - 0.771i)T \)
5 \( 1 + (-0.860 + 0.508i)T \)
7 \( 1 + (-0.972 - 0.233i)T \)
11 \( 1 + (-0.556 + 0.831i)T \)
13 \( 1 + (0.606 - 0.794i)T \)
17 \( 1 + (0.975 + 0.221i)T \)
19 \( 1 + (0.827 + 0.561i)T \)
29 \( 1 + (-0.0186 + 0.999i)T \)
31 \( 1 + (0.980 - 0.197i)T \)
37 \( 1 + (0.323 - 0.946i)T \)
41 \( 1 + (0.566 + 0.824i)T \)
43 \( 1 + (0.275 + 0.961i)T \)
47 \( 1 + (-0.334 + 0.942i)T \)
53 \( 1 + (-0.873 - 0.487i)T \)
59 \( 1 + (-0.166 - 0.985i)T \)
61 \( 1 + (0.626 + 0.779i)T \)
67 \( 1 + (-0.986 - 0.160i)T \)
71 \( 1 + (-0.993 + 0.111i)T \)
73 \( 1 + (-0.0186 - 0.999i)T \)
79 \( 1 + (0.751 + 0.659i)T \)
83 \( 1 + (-0.449 + 0.893i)T \)
89 \( 1 + (0.130 - 0.991i)T \)
97 \( 1 + (0.922 - 0.386i)T \)
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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

−23.3773863744316383962972000854, −22.66806584740133847917038398681, −21.901891526138497751093252746777, −21.03784791705481479521974827777, −20.49614426314136238721861492777, −19.35843135319207511203438450400, −18.70164852397699364332758226168, −17.01832877953107000994302938785, −16.277267869156520806014416710717, −15.86140734590290816868506534192, −15.260108486609896664631761301187, −13.95015304104441672550562761042, −13.14208685385631508356873484511, −11.992634077220245721017159654352, −11.70661920576403477947981243383, −10.6858285008395254469566067941, −9.69486970754558654314110938650, −8.5818779422630197096760000064, −7.30895837624502695779755949872, −6.219457406790076709542571304547, −5.4924406793686160184210338701, −4.5371394861833709097310640351, −3.61759487332581776516219289300, −2.957832729871523733685641639295, −0.839059486957587562953874570766, 1.15602554758108282097315726176, 2.745379567485259349029693817107, 3.45164227817591907145915179961, 4.68495867465471330514221795037, 5.78750483438896607673928525863, 6.500824960510513935670126273333, 7.53794790235873563392425249226, 7.86622028400374898424566400398, 10.02504336774320784590359657231, 10.74078815375798926249627341235, 11.6487250180050393146256345304, 12.59591963000001255054652569089, 12.84965723615365777802081568226, 14.0083546421656039591876438911, 14.90307672053516351850576946905, 16.07410022260627341295574685458, 16.22393452570039061581495615363, 17.63967422647800571669989040525, 18.56254161244495714036748599651, 19.42188182305354702190152436220, 20.072932782390635598499895801030, 21.060651000310468883208141808328, 22.481805380401528180286981455358, 22.684353446446534887135508811177, 23.33443506434696784883140271947

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