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Properties

Label 1-23e2-529.142-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.598 - 0.801i$
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  + (−0.426 + 0.904i)2-s + (0.664 − 0.747i)3-s + (−0.635 − 0.771i)4-s + (−0.743 − 0.668i)5-s + (0.392 + 0.919i)6-s + (−0.966 − 0.257i)7-s + (0.969 − 0.245i)8-s + (−0.117 − 0.993i)9-s + (0.922 − 0.386i)10-s + (0.606 − 0.794i)11-s + (−0.999 − 0.0372i)12-s + (0.566 + 0.824i)13-s + (0.645 − 0.763i)14-s + (−0.993 + 0.111i)15-s + (−0.191 + 0.981i)16-s + (0.735 − 0.678i)17-s + ⋯
L(s)  = 1  + (−0.426 + 0.904i)2-s + (0.664 − 0.747i)3-s + (−0.635 − 0.771i)4-s + (−0.743 − 0.668i)5-s + (0.392 + 0.919i)6-s + (−0.966 − 0.257i)7-s + (0.969 − 0.245i)8-s + (−0.117 − 0.993i)9-s + (0.922 − 0.386i)10-s + (0.606 − 0.794i)11-s + (−0.999 − 0.0372i)12-s + (0.566 + 0.824i)13-s + (0.645 − 0.763i)14-s + (−0.993 + 0.111i)15-s + (−0.191 + 0.981i)16-s + (0.735 − 0.678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.598 - 0.801i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.598 - 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3035254080 - 0.6056515603i\)
\(L(\frac12)\) \(\approx\) \(0.3035254080 - 0.6056515603i\)
\(L(1)\) \(\approx\) \(0.7255972184 - 0.1774622992i\)
\(L(1)\) \(\approx\) \(0.7255972184 - 0.1774622992i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.426 + 0.904i)T \)
3 \( 1 + (0.664 - 0.747i)T \)
5 \( 1 + (-0.743 - 0.668i)T \)
7 \( 1 + (-0.966 - 0.257i)T \)
11 \( 1 + (0.606 - 0.794i)T \)
13 \( 1 + (0.566 + 0.824i)T \)
17 \( 1 + (0.735 - 0.678i)T \)
19 \( 1 + (-0.404 - 0.914i)T \)
29 \( 1 + (-0.998 + 0.0620i)T \)
31 \( 1 + (-0.926 + 0.375i)T \)
37 \( 1 + (0.998 - 0.0496i)T \)
41 \( 1 + (-0.996 + 0.0868i)T \)
43 \( 1 + (-0.596 - 0.802i)T \)
47 \( 1 + (-0.576 - 0.816i)T \)
53 \( 1 + (0.922 + 0.386i)T \)
59 \( 1 + (-0.847 - 0.530i)T \)
61 \( 1 + (-0.986 - 0.160i)T \)
67 \( 1 + (-0.873 - 0.487i)T \)
71 \( 1 + (0.931 + 0.363i)T \)
73 \( 1 + (-0.998 - 0.0620i)T \)
79 \( 1 + (-0.215 + 0.976i)T \)
83 \( 1 + (-0.0186 + 0.999i)T \)
89 \( 1 + (-0.907 + 0.421i)T \)
97 \( 1 + (0.717 - 0.696i)T \)
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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

−23.26912412828875415617417791902, −22.65316893575823766107489644222, −22.09654986808351506124982323939, −21.14669721918127034378675850842, −20.16864716830456865448928527419, −19.759303500416495968104429598429, −18.903961239353279846720133021027, −18.30356970545566683655934484089, −16.890739847480895555930438749946, −16.25042358392340665174215054103, −15.09316046599197511532228364985, −14.618635994124693469231053446276, −13.27523659297167329711899154152, −12.54118494869089533121783460015, −11.54565229326249967873935959535, −10.51656086480435122295584771529, −10.00584141275445134731603937300, −9.12769897782961503261590413114, −8.141869289744466022495713795973, −7.41666661824283702513564109156, −5.9279431682328113041953933759, −4.33775093472307989294840226198, −3.57051987615051780336358741510, −3.05442246416524137038004944824, −1.78150753713107424558341816066, 0.40001613652922966402632699128, 1.49163043151927070677243835710, 3.324594150398551938193137151401, 4.132602781427822431339530611102, 5.57598154127847236645008868179, 6.65817381012992383802595753569, 7.20692762937029811299350244955, 8.25334998386344615163813710781, 9.02243211841849447367901289594, 9.49319397209237666219391038601, 11.08806208573630752501965864431, 12.13426132469780833703640942709, 13.2484734476602994824612089443, 13.68563072051916940542739792510, 14.73901902626395267711519702725, 15.61725839375282620402753616231, 16.570713264253617855939299241500, 16.88221755326000306134402668443, 18.455859208377852757362895246759, 18.81991688463665910757590983398, 19.74777680529925321747266203662, 20.11723868210794273989550065368, 21.54631564181361368138594948949, 22.79251116818941084114549594298, 23.60046293947317124048572444908

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