Properties

Label 1-23e2-529.507-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.993 - 0.112i$
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  + (−0.990 − 0.136i)2-s + (−0.990 − 0.136i)3-s + (0.962 + 0.269i)4-s + (0.203 − 0.979i)5-s + (0.962 + 0.269i)6-s + (0.962 − 0.269i)7-s + (−0.917 − 0.398i)8-s + (0.962 + 0.269i)9-s + (−0.334 + 0.942i)10-s + (−0.775 + 0.631i)11-s + (−0.917 − 0.398i)12-s + (−0.334 − 0.942i)13-s + (−0.990 + 0.136i)14-s + (−0.334 + 0.942i)15-s + (0.854 + 0.519i)16-s + (−0.334 − 0.942i)17-s + ⋯
L(s)  = 1  + (−0.990 − 0.136i)2-s + (−0.990 − 0.136i)3-s + (0.962 + 0.269i)4-s + (0.203 − 0.979i)5-s + (0.962 + 0.269i)6-s + (0.962 − 0.269i)7-s + (−0.917 − 0.398i)8-s + (0.962 + 0.269i)9-s + (−0.334 + 0.942i)10-s + (−0.775 + 0.631i)11-s + (−0.917 − 0.398i)12-s + (−0.334 − 0.942i)13-s + (−0.990 + 0.136i)14-s + (−0.334 + 0.942i)15-s + (0.854 + 0.519i)16-s + (−0.334 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.993 - 0.112i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.993 - 0.112i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01943364914 - 0.3440926720i\)
\(L(\frac12)\) \(\approx\) \(0.01943364914 - 0.3440926720i\)
\(L(1)\) \(\approx\) \(0.4385733272 - 0.2064472159i\)
\(L(1)\) \(\approx\) \(0.4385733272 - 0.2064472159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.990 - 0.136i)T \)
3 \( 1 + (-0.990 - 0.136i)T \)
5 \( 1 + (0.203 - 0.979i)T \)
7 \( 1 + (0.962 - 0.269i)T \)
11 \( 1 + (-0.775 + 0.631i)T \)
13 \( 1 + (-0.334 - 0.942i)T \)
17 \( 1 + (-0.334 - 0.942i)T \)
19 \( 1 + (-0.990 - 0.136i)T \)
29 \( 1 + (-0.775 + 0.631i)T \)
31 \( 1 + (0.460 - 0.887i)T \)
37 \( 1 + (0.854 - 0.519i)T \)
41 \( 1 + (-0.576 + 0.816i)T \)
43 \( 1 + (0.682 + 0.730i)T \)
47 \( 1 + (0.460 + 0.887i)T \)
53 \( 1 + (-0.334 - 0.942i)T \)
59 \( 1 + (-0.990 + 0.136i)T \)
61 \( 1 + (0.203 - 0.979i)T \)
67 \( 1 + (-0.775 + 0.631i)T \)
71 \( 1 + (-0.576 - 0.816i)T \)
73 \( 1 + (-0.775 - 0.631i)T \)
79 \( 1 + (0.682 + 0.730i)T \)
83 \( 1 + (0.203 - 0.979i)T \)
89 \( 1 + (-0.0682 - 0.997i)T \)
97 \( 1 + (-0.576 - 0.816i)T \)
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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

−23.84252405655883481467320679997, −23.384050096373350545225851937305, −21.85400830230033657320435726453, −21.5372283957261433380467119011, −20.69172793417509440237697437939, −19.07344586301653721535598918489, −18.81713559966569652085890721279, −17.89844089013401041738502022419, −17.27509652862003339135321994059, −16.56221873914736015915369270019, −15.37822572567574874206390806718, −14.96178283095664871492268062001, −13.7484837754903424766328866220, −12.27870149933714194411926834849, −11.44334111840653743103075323302, −10.74799830097295257477032656763, −10.319501268747604320964428059167, −9.06371219541842457692059022321, −8.027301088050840842935170688343, −7.08340306970766507985407756324, −6.222772049811906845285361936, −5.494682098407870944690645413664, −4.126291804043650104869810956675, −2.478191423796042932752545515144, −1.58978070592634345526940486059, 0.29053060591752841019191667208, 1.42434021815437768947018012575, 2.42608064535999162643086230770, 4.42803009886958549514456849956, 5.16398292035234809844370355729, 6.17608903051468723344634702260, 7.51759680723645594217470240397, 7.91538207497669622630826816673, 9.17386165400607247049349199766, 10.084228762960541196254498868981, 10.8902251273706836035090363433, 11.64434549826125667702082068978, 12.6202373544403249935933714326, 13.16885793638589442903345250069, 14.89933998784433016533073859154, 15.75170693339438316573083363817, 16.58911481909871626345466149702, 17.35912062182896359608887347362, 17.81857271588144255139363992049, 18.51861289239439508777388258223, 19.78405476663333069837250149851, 20.6288326925996295270011749293, 21.06103255403955354378075398354, 22.12457235641112336779574631923, 23.33744638491885125652209347465

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