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Properties

Label 1-23e2-529.151-r0-0-0
Degree $1$
Conductor $529$
Sign $0.999 + 0.0118i$
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  + (0.997 + 0.0744i)2-s + (0.879 − 0.476i)3-s + (0.988 + 0.148i)4-s + (0.0310 + 0.999i)5-s + (0.912 − 0.409i)6-s + (−0.759 − 0.650i)7-s + (0.975 + 0.221i)8-s + (0.545 − 0.837i)9-s + (−0.0434 + 0.999i)10-s + (0.980 + 0.197i)11-s + (0.940 − 0.340i)12-s + (−0.926 − 0.375i)13-s + (−0.709 − 0.704i)14-s + (0.503 + 0.863i)15-s + (0.955 + 0.293i)16-s + (0.783 + 0.621i)17-s + ⋯
L(s)  = 1  + (0.997 + 0.0744i)2-s + (0.879 − 0.476i)3-s + (0.988 + 0.148i)4-s + (0.0310 + 0.999i)5-s + (0.912 − 0.409i)6-s + (−0.759 − 0.650i)7-s + (0.975 + 0.221i)8-s + (0.545 − 0.837i)9-s + (−0.0434 + 0.999i)10-s + (0.980 + 0.197i)11-s + (0.940 − 0.340i)12-s + (−0.926 − 0.375i)13-s + (−0.709 − 0.704i)14-s + (0.503 + 0.863i)15-s + (0.955 + 0.293i)16-s + (0.783 + 0.621i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.999 + 0.0118i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 + 0.0118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.429832919 + 0.02036912018i\)
\(L(\frac12)\) \(\approx\) \(3.429832919 + 0.02036912018i\)
\(L(1)\) \(\approx\) \(2.429232683 + 0.003251332821i\)
\(L(1)\) \(\approx\) \(2.429232683 + 0.003251332821i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.997 + 0.0744i)T \)
3 \( 1 + (0.879 - 0.476i)T \)
5 \( 1 + (0.0310 + 0.999i)T \)
7 \( 1 + (-0.759 - 0.650i)T \)
11 \( 1 + (0.980 + 0.197i)T \)
13 \( 1 + (-0.926 - 0.375i)T \)
17 \( 1 + (0.783 + 0.621i)T \)
19 \( 1 + (-0.215 + 0.976i)T \)
29 \( 1 + (0.0558 - 0.998i)T \)
31 \( 1 + (0.827 - 0.561i)T \)
37 \( 1 + (-0.834 + 0.551i)T \)
41 \( 1 + (-0.972 + 0.233i)T \)
43 \( 1 + (-0.743 - 0.668i)T \)
47 \( 1 + (0.854 - 0.519i)T \)
53 \( 1 + (-0.0434 - 0.999i)T \)
59 \( 1 + (0.481 + 0.876i)T \)
61 \( 1 + (-0.896 + 0.443i)T \)
67 \( 1 + (-0.885 - 0.465i)T \)
71 \( 1 + (-0.944 + 0.329i)T \)
73 \( 1 + (0.0558 + 0.998i)T \)
79 \( 1 + (-0.556 + 0.831i)T \)
83 \( 1 + (0.984 - 0.172i)T \)
89 \( 1 + (-0.381 + 0.924i)T \)
97 \( 1 + (0.369 - 0.929i)T \)
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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

−23.53634235443156664299195145582, −22.31196178813722811927418755432, −21.805946547453868201086376102755, −21.11579491502153662028480119442, −20.154582994991093205631737419022, −19.6042733641417589814044853544, −18.98223580520702986846007643437, −17.1502289967210607551688400337, −16.3388579159865834989424849827, −15.76679735147176202447762996851, −14.853306113147998834856965550043, −14.05218131017094267102982828035, −13.295263203926660123710051248402, −12.331957027152322665820431414138, −11.835195651719832575370547980083, −10.361072424937519237408722178866, −9.37468327645961889485758512701, −8.83078450990552505305273240478, −7.50321270761708457216223674139, −6.51048173166437952771584258478, −5.22442359820085533006149978828, −4.62175212644591587501356687084, −3.51128224692194224008833672909, −2.69124744513098680423475080266, −1.52635119962486615436630009841, 1.58138763908863885308232359638, 2.68548604502081037934096210405, 3.523282024423113263414779288139, 4.138169833391755815671482899836, 5.89166770920186631944984779227, 6.69736998328072961322826109145, 7.30507280812585929293390812548, 8.19615803520552100392093440562, 9.94135427287415058735198451356, 10.21715809107124090076462259128, 11.81522600777228988555572867291, 12.36466965965100313423404736041, 13.4778468193063147006466964714, 14.008143214080152730255028860167, 14.89477861206799695764985610912, 15.24567390609203620916020336976, 16.68015239784225572629701593740, 17.39315402358414292114629671453, 18.93377207438911794415137577545, 19.329160071777728393822883343268, 20.12173908998507068556968623229, 20.98457540381654443898049260716, 21.99280932267867137837581143986, 22.70699777955066689304250213610, 23.326105083168653775822335876253

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