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Properties

Label 1-23e2-529.234-r0-0-0
Degree $1$
Conductor $529$
Sign $0.569 + 0.822i$
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  + (0.299 − 0.954i)2-s + (−0.556 + 0.831i)3-s + (−0.820 − 0.571i)4-s + (0.503 − 0.863i)5-s + (0.626 + 0.779i)6-s + (−0.860 − 0.508i)7-s + (−0.791 + 0.611i)8-s + (−0.381 − 0.924i)9-s + (−0.673 − 0.739i)10-s + (−0.972 + 0.233i)11-s + (0.931 − 0.363i)12-s + (−0.966 + 0.257i)13-s + (−0.743 + 0.668i)14-s + (0.437 + 0.899i)15-s + (0.346 + 0.938i)16-s + (0.392 + 0.919i)17-s + ⋯
L(s)  = 1  + (0.299 − 0.954i)2-s + (−0.556 + 0.831i)3-s + (−0.820 − 0.571i)4-s + (0.503 − 0.863i)5-s + (0.626 + 0.779i)6-s + (−0.860 − 0.508i)7-s + (−0.791 + 0.611i)8-s + (−0.381 − 0.924i)9-s + (−0.673 − 0.739i)10-s + (−0.972 + 0.233i)11-s + (0.931 − 0.363i)12-s + (−0.966 + 0.257i)13-s + (−0.743 + 0.668i)14-s + (0.437 + 0.899i)15-s + (0.346 + 0.938i)16-s + (0.392 + 0.919i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.569 + 0.822i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.569 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3965406825 + 0.2077846505i\)
\(L(\frac12)\) \(\approx\) \(0.3965406825 + 0.2077846505i\)
\(L(1)\) \(\approx\) \(0.6731607613 - 0.2118975594i\)
\(L(1)\) \(\approx\) \(0.6731607613 - 0.2118975594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.299 - 0.954i)T \)
3 \( 1 + (-0.556 + 0.831i)T \)
5 \( 1 + (0.503 - 0.863i)T \)
7 \( 1 + (-0.860 - 0.508i)T \)
11 \( 1 + (-0.972 + 0.233i)T \)
13 \( 1 + (-0.966 + 0.257i)T \)
17 \( 1 + (0.392 + 0.919i)T \)
19 \( 1 + (0.524 + 0.851i)T \)
29 \( 1 + (0.813 + 0.581i)T \)
31 \( 1 + (-0.759 - 0.650i)T \)
37 \( 1 + (0.879 + 0.476i)T \)
41 \( 1 + (0.645 + 0.763i)T \)
43 \( 1 + (-0.993 - 0.111i)T \)
47 \( 1 + (-0.990 + 0.136i)T \)
53 \( 1 + (-0.673 + 0.739i)T \)
59 \( 1 + (0.767 + 0.640i)T \)
61 \( 1 + (-0.0434 - 0.999i)T \)
67 \( 1 + (0.369 + 0.929i)T \)
71 \( 1 + (-0.834 + 0.551i)T \)
73 \( 1 + (0.813 - 0.581i)T \)
79 \( 1 + (0.566 + 0.824i)T \)
83 \( 1 + (-0.982 + 0.185i)T \)
89 \( 1 + (-0.358 - 0.933i)T \)
97 \( 1 + (0.154 + 0.987i)T \)
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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

−23.28958279953011370131373942039, −22.66001602557373366057574347212, −22.05479539046211548714199944872, −21.33299286787246730646923884912, −19.64016083340392857300184942967, −18.79800644826484725530060453826, −18.08296813321708683748078363763, −17.633424033198151777299364155, −16.467728295648059439028118186229, −15.828453846670429472435504119152, −14.794911413452809994743528194892, −13.85933331373115305093094684504, −13.188243304135843851335991563785, −12.443262215242581608771035969261, −11.451510001195394448418523987953, −10.16786720373154399216544227694, −9.349085440534339831911907479635, −7.9842989918540613005138335672, −7.174518267388174209195115362402, −6.57207322228981357223292719738, −5.57963129450564348853473117613, −5.05368647047468288676498483543, −3.09485123322878227436743376433, −2.50630952926975736467093337681, −0.24528245180916536285887338578, 1.226010303870498048236561471942, 2.68978610822609034104112206794, 3.77782949244694775262648643216, 4.68627263979998743430752407124, 5.430775131865489816667746683113, 6.30139088214924938149119062348, 8.031191672848352601526139146090, 9.27964479396773482808843035820, 9.97631016278843739056549395840, 10.3059530475165110267151463725, 11.54000978482213760671111865486, 12.57564138950157269842959956915, 12.899050022833179818794197424086, 14.097183321966900758950524526738, 14.99594581201971500318987627847, 16.176969682171170820214774574695, 16.810780649111932866914299347898, 17.65229424137986338237649568956, 18.62041450615653912004417866709, 19.84170548162836165431749909211, 20.309232169902706998544077072660, 21.233623138605670625705469705496, 21.74149900673317468906861740642, 22.58973776038552913126952196003, 23.462930815981686208108816725695

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