Properties

Label 1-23e2-529.353-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: $\chi_{529} (353, \cdot )$
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

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

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