Properties

Label 1-23e2-529.100-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.830 + 0.557i$
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  + (0.783 − 0.621i)2-s + (−0.239 − 0.970i)3-s + (0.227 − 0.973i)4-s + (0.275 − 0.961i)5-s + (−0.791 − 0.611i)6-s + (−0.996 + 0.0868i)7-s + (−0.426 − 0.904i)8-s + (−0.885 + 0.465i)9-s + (−0.381 − 0.924i)10-s + (−0.215 − 0.976i)11-s + (−0.999 + 0.0124i)12-s + (0.948 − 0.317i)13-s + (−0.726 + 0.687i)14-s + (−0.999 − 0.0372i)15-s + (−0.896 − 0.443i)16-s + (0.969 + 0.245i)17-s + ⋯
L(s)  = 1  + (0.783 − 0.621i)2-s + (−0.239 − 0.970i)3-s + (0.227 − 0.973i)4-s + (0.275 − 0.961i)5-s + (−0.791 − 0.611i)6-s + (−0.996 + 0.0868i)7-s + (−0.426 − 0.904i)8-s + (−0.885 + 0.465i)9-s + (−0.381 − 0.924i)10-s + (−0.215 − 0.976i)11-s + (−0.999 + 0.0124i)12-s + (0.948 − 0.317i)13-s + (−0.726 + 0.687i)14-s + (−0.999 − 0.0372i)15-s + (−0.896 − 0.443i)16-s + (0.969 + 0.245i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.830 + 0.557i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.830 + 0.557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4445424616 - 1.459872335i\)
\(L(\frac12)\) \(\approx\) \(-0.4445424616 - 1.459872335i\)
\(L(1)\) \(\approx\) \(0.6588679443 - 1.125993508i\)
\(L(1)\) \(\approx\) \(0.6588679443 - 1.125993508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.783 - 0.621i)T \)
3 \( 1 + (-0.239 - 0.970i)T \)
5 \( 1 + (0.275 - 0.961i)T \)
7 \( 1 + (-0.996 + 0.0868i)T \)
11 \( 1 + (-0.215 - 0.976i)T \)
13 \( 1 + (0.948 - 0.317i)T \)
17 \( 1 + (0.969 + 0.245i)T \)
19 \( 1 + (-0.926 + 0.375i)T \)
29 \( 1 + (0.481 + 0.876i)T \)
31 \( 1 + (0.606 - 0.794i)T \)
37 \( 1 + (-0.514 + 0.857i)T \)
41 \( 1 + (0.524 - 0.851i)T \)
43 \( 1 + (-0.952 + 0.305i)T \)
47 \( 1 + (0.203 - 0.979i)T \)
53 \( 1 + (-0.381 + 0.924i)T \)
59 \( 1 + (-0.982 + 0.185i)T \)
61 \( 1 + (0.545 + 0.837i)T \)
67 \( 1 + (0.346 - 0.938i)T \)
71 \( 1 + (0.992 - 0.123i)T \)
73 \( 1 + (0.481 - 0.876i)T \)
79 \( 1 + (0.827 - 0.561i)T \)
83 \( 1 + (0.00620 + 0.999i)T \)
89 \( 1 + (0.369 + 0.929i)T \)
97 \( 1 + (-0.263 - 0.964i)T \)
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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

−23.3226916707246069180924557440, −23.154022540814699859329875284, −22.5266401297503995734813497950, −21.501691529026593605425058077594, −21.12631458865914234300023965041, −20.03949236647178586562168048441, −18.882725967326779319427467746382, −17.733206621833113105492741069041, −17.0644384432506206885808653467, −15.98550140029041315570511909680, −15.59117997695741348684194044434, −14.66606086579888335441639123847, −13.97999735625024676789945447125, −12.96259930992269670061799206092, −11.98626997497434921123664585614, −10.97966764458115365944366958516, −10.120876749597800008778763242256, −9.28149777640530401939427319839, −8.02271783449127293769014198629, −6.72249481556175766363991443943, −6.30283953280845235487091356742, −5.24231674966487720433676213424, −4.12545191159268821879811345708, −3.36397337698642939411567218302, −2.47083326794966682857089733997, 0.63827787709188252966135999691, 1.582373540276344881575942197381, 2.85607960731030527057003546470, 3.770336661227176111130872847187, 5.239315404857244542865078047175, 5.9596682299892435771300445715, 6.52259915898378650351878974866, 8.1036779728171161015785107577, 8.92751550153815077865663375550, 10.15483331946157166078414661766, 11.013112805293942033600902915837, 12.22636898538020587057580470354, 12.541786839712868963054875574633, 13.55719358282584333075277234681, 13.748423269024802120768488953106, 15.238014719274753464547369439656, 16.31704971654576698838474002342, 16.86836491425805929347412861752, 18.27300514830887317518918312275, 18.9713387469029349788841683278, 19.5870363381453868402911406841, 20.52170416702517253784356697429, 21.29144214483252547840270708370, 22.18169803403466125624916774055, 23.2224143899383900052593655736

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