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Properties

Label 1-23e2-529.16-r0-0-0
Degree $1$
Conductor $529$
Sign $0.252 + 0.967i$
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  + (0.275 + 0.961i)2-s + (−0.287 + 0.957i)3-s + (−0.847 + 0.530i)4-s + (−0.873 − 0.487i)5-s + (−0.999 − 0.0124i)6-s + (0.955 − 0.293i)7-s + (−0.743 − 0.668i)8-s + (−0.834 − 0.551i)9-s + (0.227 − 0.973i)10-s + (0.735 − 0.678i)11-s + (−0.263 − 0.964i)12-s + (−0.791 + 0.611i)13-s + (0.545 + 0.837i)14-s + (0.717 − 0.696i)15-s + (0.437 − 0.899i)16-s + (0.586 − 0.809i)17-s + ⋯
L(s)  = 1  + (0.275 + 0.961i)2-s + (−0.287 + 0.957i)3-s + (−0.847 + 0.530i)4-s + (−0.873 − 0.487i)5-s + (−0.999 − 0.0124i)6-s + (0.955 − 0.293i)7-s + (−0.743 − 0.668i)8-s + (−0.834 − 0.551i)9-s + (0.227 − 0.973i)10-s + (0.735 − 0.678i)11-s + (−0.263 − 0.964i)12-s + (−0.791 + 0.611i)13-s + (0.545 + 0.837i)14-s + (0.717 − 0.696i)15-s + (0.437 − 0.899i)16-s + (0.586 − 0.809i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9099649765 + 0.7027068192i\)
\(L(\frac12)\) \(\approx\) \(0.9099649765 + 0.7027068192i\)
\(L(1)\) \(\approx\) \(0.8117980485 + 0.5431318709i\)
\(L(1)\) \(\approx\) \(0.8117980485 + 0.5431318709i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.275 + 0.961i)T \)
3 \( 1 + (-0.287 + 0.957i)T \)
5 \( 1 + (-0.873 - 0.487i)T \)
7 \( 1 + (0.955 - 0.293i)T \)
11 \( 1 + (0.735 - 0.678i)T \)
13 \( 1 + (-0.791 + 0.611i)T \)
17 \( 1 + (0.586 - 0.809i)T \)
19 \( 1 + (0.912 - 0.409i)T \)
29 \( 1 + (0.566 + 0.824i)T \)
31 \( 1 + (-0.616 + 0.787i)T \)
37 \( 1 + (-0.166 - 0.985i)T \)
41 \( 1 + (0.995 + 0.0991i)T \)
43 \( 1 + (-0.381 - 0.924i)T \)
47 \( 1 + (0.460 - 0.887i)T \)
53 \( 1 + (0.227 + 0.973i)T \)
59 \( 1 + (-0.759 - 0.650i)T \)
61 \( 1 + (-0.806 + 0.591i)T \)
67 \( 1 + (-0.514 + 0.857i)T \)
71 \( 1 + (0.890 + 0.454i)T \)
73 \( 1 + (0.566 - 0.824i)T \)
79 \( 1 + (0.969 - 0.245i)T \)
83 \( 1 + (0.606 - 0.794i)T \)
89 \( 1 + (0.879 + 0.476i)T \)
97 \( 1 + (-0.239 - 0.970i)T \)
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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

−23.03389338869133557764306402852, −22.63700702962183701217017509509, −21.7850763314307799903429056476, −20.587176800937294910474998211808, −19.82430245133865807789337849247, −19.227717400827975079086957283710, −18.36966264495895340792214427225, −17.74243441387894780665359099058, −16.91519312906151721717452652816, −15.17300354456257875329078610620, −14.63123976151618673125790147876, −13.86412942465045047214098713383, −12.53741424725277658656453341641, −12.114382005687003254257522657764, −11.45274202701952725862092894857, −10.62775855727907872985525384366, −9.49481156544139258060045831576, −8.07464493927072895547498923083, −7.70936742316583193374764946349, −6.32291913020550322140112958158, −5.25521510109026388502870881180, −4.28557630371268838434076384085, −3.07965582788221179458594048140, −2.053311825748011732540315637847, −1.02280826876320367548258915395, 0.79345951001233645090161827424, 3.23635095028473504251938304441, 4.09211268357242968270469443125, 4.904664925401306293641126416991, 5.4615002210063636414234988069, 6.949601818411550345705031807673, 7.71536014717445199052894381071, 8.85752166166918951003280290433, 9.27022274183435476598466449242, 10.72845748073594279590406042348, 11.78637128173281150686124371754, 12.186313052137449363329136944486, 13.91377558860039974091590628185, 14.363954983124512531206189209722, 15.221144526906755093089452863035, 16.19341687948405138014885421266, 16.562591694937900207188792578269, 17.3786433961075228225277962479, 18.315527020025965339738140549091, 19.59903401392180914812307062943, 20.44120854992903299151977395194, 21.447551958576114008702039012943, 21.988215067953030111048110175435, 23.007455492318050031874240688017, 23.64166227064963736102368669232

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