Properties

Label 1-23e2-529.269-r0-0-0
Degree $1$
Conductor $529$
Sign $0.823 - 0.566i$
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  + (−0.820 − 0.571i)2-s + (−0.381 − 0.924i)3-s + (0.346 + 0.938i)4-s + (−0.492 − 0.870i)5-s + (−0.215 + 0.976i)6-s + (0.481 + 0.876i)7-s + (0.251 − 0.967i)8-s + (−0.709 + 0.704i)9-s + (−0.0929 + 0.995i)10-s + (0.890 − 0.454i)11-s + (0.735 − 0.678i)12-s + (0.867 − 0.498i)13-s + (0.105 − 0.994i)14-s + (−0.616 + 0.787i)15-s + (−0.759 + 0.650i)16-s + (−0.691 + 0.722i)17-s + ⋯
L(s)  = 1  + (−0.820 − 0.571i)2-s + (−0.381 − 0.924i)3-s + (0.346 + 0.938i)4-s + (−0.492 − 0.870i)5-s + (−0.215 + 0.976i)6-s + (0.481 + 0.876i)7-s + (0.251 − 0.967i)8-s + (−0.709 + 0.704i)9-s + (−0.0929 + 0.995i)10-s + (0.890 − 0.454i)11-s + (0.735 − 0.678i)12-s + (0.867 − 0.498i)13-s + (0.105 − 0.994i)14-s + (−0.616 + 0.787i)15-s + (−0.759 + 0.650i)16-s + (−0.691 + 0.722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.823 - 0.566i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.823 - 0.566i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6937915804 - 0.2155700476i\)
\(L(\frac12)\) \(\approx\) \(0.6937915804 - 0.2155700476i\)
\(L(1)\) \(\approx\) \(0.6062199069 - 0.2586007429i\)
\(L(1)\) \(\approx\) \(0.6062199069 - 0.2586007429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.820 - 0.571i)T \)
3 \( 1 + (-0.381 - 0.924i)T \)
5 \( 1 + (-0.492 - 0.870i)T \)
7 \( 1 + (0.481 + 0.876i)T \)
11 \( 1 + (0.890 - 0.454i)T \)
13 \( 1 + (0.867 - 0.498i)T \)
17 \( 1 + (-0.691 + 0.722i)T \)
19 \( 1 + (-0.449 + 0.893i)T \)
29 \( 1 + (0.323 + 0.946i)T \)
31 \( 1 + (0.154 + 0.987i)T \)
37 \( 1 + (0.545 + 0.837i)T \)
41 \( 1 + (-0.166 + 0.985i)T \)
43 \( 1 + (0.975 + 0.221i)T \)
47 \( 1 + (0.962 - 0.269i)T \)
53 \( 1 + (-0.0929 - 0.995i)T \)
59 \( 1 + (0.179 + 0.983i)T \)
61 \( 1 + (-0.996 + 0.0868i)T \)
67 \( 1 + (-0.726 + 0.687i)T \)
71 \( 1 + (0.392 - 0.919i)T \)
73 \( 1 + (0.323 - 0.946i)T \)
79 \( 1 + (-0.358 + 0.933i)T \)
83 \( 1 + (0.931 - 0.363i)T \)
89 \( 1 + (-0.743 + 0.668i)T \)
97 \( 1 + (-0.952 + 0.305i)T \)
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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

−23.43854459947335825586493921483, −22.92674525345300430684986859663, −22.08081445886861492738179097025, −20.845753444510616088796274347577, −20.13271229199011981342746231517, −19.34943776262247635840638907929, −18.287434896929643543982291382758, −17.44857342163559460181165430445, −16.94895073129524501138198441502, −15.80407153415044854939474407480, −15.38211933640275759256571309705, −14.39495968183610871634754819471, −13.813887114469868569982200252102, −11.74761448267808983371861649615, −11.123334045340562760171823388592, −10.630448550142242825459994293554, −9.54979696903695964138972896910, −8.83142139576558412525263028357, −7.60307572721437897127228893765, −6.77948915198820338542148518646, −6.04508409090836299943885348132, −4.53246200419845961321127258776, −3.99066849293037108563475813979, −2.361323687592709320639813233264, −0.65706951711638092291336887720, 1.122410811179632127721716555517, 1.71548445089184164226216965598, 3.12508533337223865014380236116, 4.36759008481799519783280348541, 5.74945417490463407985042486540, 6.61801268287546040830261276648, 7.96940775452902010517217878551, 8.47196379014622559157283572652, 9.04317139794606961283862456914, 10.65799294714048248595050519429, 11.38392465285454435594999446798, 12.17298732994682877410271890242, 12.65397600119900360997572572229, 13.62220639134040892382865856497, 15.0204553484444855783904250778, 16.16177909528328561802560554441, 16.799224600112391275783910984548, 17.712105510813238502068697907375, 18.31976104521873442057738998829, 19.25819547325507347905099888791, 19.758650603227025480861588152239, 20.70152595945258138669749728541, 21.60096976008661644678509592648, 22.46225145382449437561602872485, 23.58118497040415095241660681473

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