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Properties

Label 1-23e2-529.128-r0-0-0
Degree $1$
Conductor $529$
Sign $0.920 + 0.390i$
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  + (−0.635 + 0.771i)2-s + (−0.117 + 0.993i)3-s + (−0.191 − 0.981i)4-s + (0.105 − 0.994i)5-s + (−0.691 − 0.722i)6-s + (0.867 − 0.498i)7-s + (0.879 + 0.476i)8-s + (−0.972 − 0.233i)9-s + (0.700 + 0.713i)10-s + (−0.263 + 0.964i)11-s + (0.997 − 0.0744i)12-s + (−0.358 − 0.933i)13-s + (−0.166 + 0.985i)14-s + (0.975 + 0.221i)15-s + (−0.926 + 0.375i)16-s + (0.0806 + 0.996i)17-s + ⋯
L(s)  = 1  + (−0.635 + 0.771i)2-s + (−0.117 + 0.993i)3-s + (−0.191 − 0.981i)4-s + (0.105 − 0.994i)5-s + (−0.691 − 0.722i)6-s + (0.867 − 0.498i)7-s + (0.879 + 0.476i)8-s + (−0.972 − 0.233i)9-s + (0.700 + 0.713i)10-s + (−0.263 + 0.964i)11-s + (0.997 − 0.0744i)12-s + (−0.358 − 0.933i)13-s + (−0.166 + 0.985i)14-s + (0.975 + 0.221i)15-s + (−0.926 + 0.375i)16-s + (0.0806 + 0.996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8962555551 + 0.1824614344i\)
\(L(\frac12)\) \(\approx\) \(0.8962555551 + 0.1824614344i\)
\(L(1)\) \(\approx\) \(0.7511731809 + 0.2450369291i\)
\(L(1)\) \(\approx\) \(0.7511731809 + 0.2450369291i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.635 + 0.771i)T \)
3 \( 1 + (-0.117 + 0.993i)T \)
5 \( 1 + (0.105 - 0.994i)T \)
7 \( 1 + (0.867 - 0.498i)T \)
11 \( 1 + (-0.263 + 0.964i)T \)
13 \( 1 + (-0.358 - 0.933i)T \)
17 \( 1 + (0.0806 + 0.996i)T \)
19 \( 1 + (-0.673 - 0.739i)T \)
29 \( 1 + (0.992 + 0.123i)T \)
31 \( 1 + (0.717 + 0.696i)T \)
37 \( 1 + (0.995 + 0.0991i)T \)
41 \( 1 + (0.984 + 0.172i)T \)
43 \( 1 + (-0.287 - 0.957i)T \)
47 \( 1 + (-0.334 - 0.942i)T \)
53 \( 1 + (0.700 - 0.713i)T \)
59 \( 1 + (0.437 - 0.899i)T \)
61 \( 1 + (0.948 - 0.317i)T \)
67 \( 1 + (0.524 - 0.851i)T \)
71 \( 1 + (0.735 - 0.678i)T \)
73 \( 1 + (0.992 - 0.123i)T \)
79 \( 1 + (-0.907 + 0.421i)T \)
83 \( 1 + (-0.999 + 0.0372i)T \)
89 \( 1 + (0.645 + 0.763i)T \)
97 \( 1 + (0.0310 + 0.999i)T \)
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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

−23.32322161808109009119292557576, −22.52366461810551411971970632731, −21.51393706324341143857525731736, −21.06408194494534314786462837971, −19.74754172047036745808092455630, −18.92709413251261025731256694847, −18.57803116524525754587987964334, −17.85919727454622919194107911610, −17.053403502755588653755675168624, −16.03955929074678542399909168329, −14.48952673122192658832875714904, −13.996246303957750902602491722934, −12.999953754690228028656688784119, −11.76283665054414371082363957706, −11.52730399811366510511780952319, −10.64826396157444841547456408046, −9.468395255276740277240439991834, −8.37561349987238055663761321420, −7.760079228926735596319132597541, −6.785236411490966304198100087824, −5.7887082193469785466099932445, −4.30619106748222286498914546402, −2.76053775717794283300065026416, −2.36673627831838615060444587078, −1.10340909016992615281716774881, 0.739338912686677008277942338971, 2.15485128076800382021665888349, 4.13647803170817660435167378895, 4.86834846067235747134575489903, 5.427281493917568932765526277307, 6.72631545448464264834219783895, 8.11584771733004194765885898471, 8.408757801658637978899478192126, 9.61203937374292053205997752469, 10.26243656353357719282336838087, 11.02312324893250637345025850699, 12.32222654982401604965014022800, 13.4403880967051147822450652206, 14.578079446366208676472837933804, 15.18585104127803882156288395354, 15.89785271465140844267270646713, 16.93528532314761577199921749355, 17.4133019418767339179465588764, 17.91592207105537878321649033913, 19.69370350789839068349895039694, 20.02344401129630595379431304979, 20.94105108194259272224647305184, 21.737803218731544774047720010307, 23.07549050487310586795530243883, 23.510826851033738834650340643653

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