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Properties

Label 1-23e2-529.514-r0-0-0
Degree $1$
Conductor $529$
Sign $0.573 - 0.819i$
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  + (−0.191 − 0.981i)2-s + (−0.972 − 0.233i)3-s + (−0.926 + 0.375i)4-s + (−0.977 − 0.209i)5-s + (−0.0434 + 0.999i)6-s + (0.503 − 0.863i)7-s + (0.545 + 0.837i)8-s + (0.890 + 0.454i)9-s + (−0.0186 + 0.999i)10-s + (−0.860 − 0.508i)11-s + (0.988 − 0.148i)12-s + (−0.743 + 0.668i)13-s + (−0.944 − 0.329i)14-s + (0.901 + 0.432i)15-s + (0.717 − 0.696i)16-s + (−0.986 + 0.160i)17-s + ⋯
L(s)  = 1  + (−0.191 − 0.981i)2-s + (−0.972 − 0.233i)3-s + (−0.926 + 0.375i)4-s + (−0.977 − 0.209i)5-s + (−0.0434 + 0.999i)6-s + (0.503 − 0.863i)7-s + (0.545 + 0.837i)8-s + (0.890 + 0.454i)9-s + (−0.0186 + 0.999i)10-s + (−0.860 − 0.508i)11-s + (0.988 − 0.148i)12-s + (−0.743 + 0.668i)13-s + (−0.944 − 0.329i)14-s + (0.901 + 0.432i)15-s + (0.717 − 0.696i)16-s + (−0.986 + 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.573 - 0.819i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.573 - 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4123958656 - 0.2146757691i\)
\(L(\frac12)\) \(\approx\) \(0.4123958656 - 0.2146757691i\)
\(L(1)\) \(\approx\) \(0.4484776340 - 0.2604088989i\)
\(L(1)\) \(\approx\) \(0.4484776340 - 0.2604088989i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.191 - 0.981i)T \)
3 \( 1 + (-0.972 - 0.233i)T \)
5 \( 1 + (-0.977 - 0.209i)T \)
7 \( 1 + (0.503 - 0.863i)T \)
11 \( 1 + (-0.860 - 0.508i)T \)
13 \( 1 + (-0.743 + 0.668i)T \)
17 \( 1 + (-0.986 + 0.160i)T \)
19 \( 1 + (-0.0929 + 0.995i)T \)
29 \( 1 + (0.969 + 0.245i)T \)
31 \( 1 + (0.0310 + 0.999i)T \)
37 \( 1 + (0.980 + 0.197i)T \)
41 \( 1 + (0.940 + 0.340i)T \)
43 \( 1 + (-0.834 + 0.551i)T \)
47 \( 1 + (-0.775 + 0.631i)T \)
53 \( 1 + (-0.0186 - 0.999i)T \)
59 \( 1 + (-0.616 - 0.787i)T \)
61 \( 1 + (0.798 - 0.601i)T \)
67 \( 1 + (-0.449 - 0.893i)T \)
71 \( 1 + (0.0806 - 0.996i)T \)
73 \( 1 + (0.969 - 0.245i)T \)
79 \( 1 + (0.645 - 0.763i)T \)
83 \( 1 + (0.997 - 0.0744i)T \)
89 \( 1 + (-0.166 + 0.985i)T \)
97 \( 1 + (-0.998 + 0.0620i)T \)
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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

−23.705577751921437949185705388677, −22.87829223825192179679383381195, −22.23460302786528466800747215311, −21.496428763081106603483872052890, −20.1252353025589349343000657068, −19.11193321421559485628688897703, −18.08383739385977083263030705727, −17.85094979633922780370343612664, −16.79957805116503647102112494695, −15.67109609903450143350602144172, −15.43844057403768083536941016493, −14.7823886537921982482812578529, −13.21628834699390921759777269686, −12.42946252137352630020275901296, −11.47152342999322880310604634626, −10.60606026201633093786364952216, −9.60740768674033359823835668473, −8.502164372286460232653532819582, −7.59765761794983225301734618352, −6.85802644784165902618159114752, −5.73705736261922238958656614513, −4.85750745955236313347971920533, −4.33196978864629899840341006629, −2.557891552070850752880272774990, −0.50173245522415591321860137065, 0.72609389233992414725832983847, 1.90373336777692419777153516868, 3.41823925688289774220364286513, 4.579327384842654050906063332600, 4.889123485126490985830479869326, 6.56651500087060532700720714022, 7.723095832359272965035924556819, 8.25621777711448535551734471951, 9.70842697738956934205957758377, 10.74349804389884888422757998612, 11.126269769571121761629853349748, 12.01784665818420346568422615, 12.73045410206071692822392377694, 13.611426949637457030709785517542, 14.67536783235410647666850466827, 16.13618753787497864768138331054, 16.6380667212309797806756988284, 17.64583620768020557710878513108, 18.300452299558611467407998547740, 19.28887352804360565560744468819, 19.820752967604193032548559926272, 20.91727713973774918834999586933, 21.572799806771467395339819778593, 22.54765600582524766801667597110, 23.43017504420924517858367107371

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