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Properties

Label 1-23e2-529.3-r0-0-0
Degree $1$
Conductor $529$
Sign $0.443 + 0.896i$
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  + (−0.449 + 0.893i)2-s + (−0.999 − 0.0372i)3-s + (−0.596 − 0.802i)4-s + (0.980 + 0.197i)5-s + (0.481 − 0.876i)6-s + (0.879 + 0.476i)7-s + (0.984 − 0.172i)8-s + (0.997 + 0.0744i)9-s + (−0.616 + 0.787i)10-s + (0.323 + 0.946i)11-s + (0.566 + 0.824i)12-s + (0.998 + 0.0496i)13-s + (−0.820 + 0.571i)14-s + (−0.972 − 0.233i)15-s + (−0.287 + 0.957i)16-s + (0.867 − 0.498i)17-s + ⋯
L(s)  = 1  + (−0.449 + 0.893i)2-s + (−0.999 − 0.0372i)3-s + (−0.596 − 0.802i)4-s + (0.980 + 0.197i)5-s + (0.481 − 0.876i)6-s + (0.879 + 0.476i)7-s + (0.984 − 0.172i)8-s + (0.997 + 0.0744i)9-s + (−0.616 + 0.787i)10-s + (0.323 + 0.946i)11-s + (0.566 + 0.824i)12-s + (0.998 + 0.0496i)13-s + (−0.820 + 0.571i)14-s + (−0.972 − 0.233i)15-s + (−0.287 + 0.957i)16-s + (0.867 − 0.498i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.443 + 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9579482213 + 0.5946525874i\)
\(L(\frac12)\) \(\approx\) \(0.9579482213 + 0.5946525874i\)
\(L(1)\) \(\approx\) \(0.8163299726 + 0.3657855033i\)
\(L(1)\) \(\approx\) \(0.8163299726 + 0.3657855033i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.449 + 0.893i)T \)
3 \( 1 + (-0.999 - 0.0372i)T \)
5 \( 1 + (0.980 + 0.197i)T \)
7 \( 1 + (0.879 + 0.476i)T \)
11 \( 1 + (0.323 + 0.946i)T \)
13 \( 1 + (0.998 + 0.0496i)T \)
17 \( 1 + (0.867 - 0.498i)T \)
19 \( 1 + (0.179 - 0.983i)T \)
29 \( 1 + (-0.0434 - 0.999i)T \)
31 \( 1 + (-0.834 + 0.551i)T \)
37 \( 1 + (0.275 - 0.961i)T \)
41 \( 1 + (-0.635 - 0.771i)T \)
43 \( 1 + (0.606 - 0.794i)T \)
47 \( 1 + (0.962 - 0.269i)T \)
53 \( 1 + (-0.616 - 0.787i)T \)
59 \( 1 + (-0.381 + 0.924i)T \)
61 \( 1 + (-0.492 - 0.870i)T \)
67 \( 1 + (0.783 + 0.621i)T \)
71 \( 1 + (-0.966 - 0.257i)T \)
73 \( 1 + (-0.0434 + 0.999i)T \)
79 \( 1 + (-0.806 + 0.591i)T \)
83 \( 1 + (-0.885 - 0.465i)T \)
89 \( 1 + (0.299 + 0.954i)T \)
97 \( 1 + (0.392 - 0.919i)T \)
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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

−23.259576743592315469280095046970, −22.18055864996352937623425850820, −21.55452879499074671006812129408, −20.93506733316639973870737112928, −20.24966747981056040796133981371, −18.68748870401818662143472772148, −18.424182929355346268613167924767, −17.44715737481090144492635255344, −16.79611391904618183976264628521, −16.29972015029522585775410234110, −14.498731373981519137265638878032, −13.67391551398868463427473839763, −12.86481444618258567146099641098, −11.92995099835638412697358968461, −11.00376608795775371361435596350, −10.55969951983555605504834767906, −9.628770971464573287860171755969, −8.57245411742530604117262132703, −7.64400568185182302012131552671, −6.21110658012500488941790071991, −5.43845668777229988529993036863, −4.33993027898077104728307233732, −3.312007602217266637715396170256, −1.488919890385612792550846196386, −1.22652992145630144823510950916, 1.12066930491620691404768424639, 2.04495753661317915725213099291, 4.23640419545270815995053117371, 5.291470150329615456411375791620, 5.71854469187803922307094444247, 6.78805581818176918844842100042, 7.48739433037513145613546192616, 8.840723295791404784344381198701, 9.603395404340841316427443156581, 10.52532712213091241834555132303, 11.319824662953017166643407146597, 12.47221741712005843130824225184, 13.5460713912242161307776259268, 14.35424900439302499813408616400, 15.302823240137523556160206930175, 16.06710185300011191702457577186, 17.18746879163759137289517714093, 17.552557107781603606691074625162, 18.29108641684270211354446513136, 18.8186345047937494516117347694, 20.38821769481187298227880472629, 21.324904151777069580436918306965, 22.14729366873306056509710850913, 22.946680057289100278515389018373, 23.67199615703514445427365631039

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