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Properties

Label 1-23e2-529.48-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.851 - 0.523i$
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  + (−0.982 − 0.185i)2-s + (0.323 − 0.946i)3-s + (0.931 + 0.363i)4-s + (−0.759 − 0.650i)5-s + (−0.492 + 0.870i)6-s + (0.980 + 0.197i)7-s + (−0.847 − 0.530i)8-s + (−0.791 − 0.611i)9-s + (0.626 + 0.779i)10-s + (0.879 + 0.476i)11-s + (0.645 − 0.763i)12-s + (−0.820 + 0.571i)13-s + (−0.926 − 0.375i)14-s + (−0.860 + 0.508i)15-s + (0.735 + 0.678i)16-s + (0.105 − 0.994i)17-s + ⋯
L(s)  = 1  + (−0.982 − 0.185i)2-s + (0.323 − 0.946i)3-s + (0.931 + 0.363i)4-s + (−0.759 − 0.650i)5-s + (−0.492 + 0.870i)6-s + (0.980 + 0.197i)7-s + (−0.847 − 0.530i)8-s + (−0.791 − 0.611i)9-s + (0.626 + 0.779i)10-s + (0.879 + 0.476i)11-s + (0.645 − 0.763i)12-s + (−0.820 + 0.571i)13-s + (−0.926 − 0.375i)14-s + (−0.860 + 0.508i)15-s + (0.735 + 0.678i)16-s + (0.105 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.851 - 0.523i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.851 - 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2076620016 - 0.7342839326i\)
\(L(\frac12)\) \(\approx\) \(0.2076620016 - 0.7342839326i\)
\(L(1)\) \(\approx\) \(0.5891629221 - 0.4059360276i\)
\(L(1)\) \(\approx\) \(0.5891629221 - 0.4059360276i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.982 - 0.185i)T \)
3 \( 1 + (0.323 - 0.946i)T \)
5 \( 1 + (-0.759 - 0.650i)T \)
7 \( 1 + (0.980 + 0.197i)T \)
11 \( 1 + (0.879 + 0.476i)T \)
13 \( 1 + (-0.820 + 0.571i)T \)
17 \( 1 + (0.105 - 0.994i)T \)
19 \( 1 + (-0.239 - 0.970i)T \)
29 \( 1 + (0.798 - 0.601i)T \)
31 \( 1 + (0.0806 - 0.996i)T \)
37 \( 1 + (-0.993 - 0.111i)T \)
41 \( 1 + (-0.556 - 0.831i)T \)
43 \( 1 + (-0.966 - 0.257i)T \)
47 \( 1 + (0.203 - 0.979i)T \)
53 \( 1 + (0.626 - 0.779i)T \)
59 \( 1 + (0.890 - 0.454i)T \)
61 \( 1 + (0.912 + 0.409i)T \)
67 \( 1 + (-0.935 + 0.352i)T \)
71 \( 1 + (-0.743 - 0.668i)T \)
73 \( 1 + (0.798 + 0.601i)T \)
79 \( 1 + (-0.635 + 0.771i)T \)
83 \( 1 + (-0.907 + 0.421i)T \)
89 \( 1 + (-0.191 + 0.981i)T \)
97 \( 1 + (-0.986 - 0.160i)T \)
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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

−23.95990298834773005793446026789, −23.04699783101998168580459473844, −21.93067498531322663166714000407, −21.2553877267561097299954050555, −20.18919605973259812877923772204, −19.653498215419445802188690691375, −18.940815912536935597222721895080, −17.78066826809236467244007535355, −17.047952226472631525230020859174, −16.27503551071396755743623733472, −15.30231707767494104567577666421, −14.63499591904293683373976419958, −14.25475846113209155973300834889, −12.15868530920655752179419430880, −11.45335597987860954497232440970, −10.49402403693683042886977880325, −10.17925649035536330338050201042, −8.70185306510088465971842643434, −8.26597571795785154497658503306, −7.36981658133301084629641706168, −6.20773936737307633310139827449, −4.99030615182296603657060327396, −3.80140226921303929793139064394, −2.8851701767443876933313966828, −1.49442822163294215427911573109, 0.554598587828228278513070281090, 1.694325912521626004218429170566, 2.53637373143767069275183062907, 4.00834859522889611999190320527, 5.2387248128329694041227779866, 6.90247813556765461609774925618, 7.24589425457734857487839382021, 8.349699979225499794882963116303, 8.83250680031861230203212523460, 9.78090878986060749165375209488, 11.45321931334765351678067954145, 11.73790033796162441457535881718, 12.39397051577982818197959295709, 13.65932140583882566195945667370, 14.775752885340006860556758165637, 15.452296882719150632523363947542, 16.75703436608505726408133999227, 17.331812269008352682926134085301, 18.084859531200594521930379338317, 19.08808369441005061086196682792, 19.59259499519524835767813348612, 20.3735462412984874390338859553, 20.99183692042490370162097540785, 22.26353282383683134155695151435, 23.54872015865671887519967714445

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