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Properties

Label 1-23e2-529.77-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.772 + 0.634i$
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  + (−0.966 + 0.257i)2-s + (−0.166 + 0.985i)3-s + (0.867 − 0.498i)4-s + (0.626 + 0.779i)5-s + (−0.0929 − 0.995i)6-s + (−0.616 + 0.787i)7-s + (−0.709 + 0.704i)8-s + (−0.944 − 0.329i)9-s + (−0.806 − 0.591i)10-s + (0.767 − 0.640i)11-s + (0.346 + 0.938i)12-s + (0.975 + 0.221i)13-s + (0.392 − 0.919i)14-s + (−0.873 + 0.487i)15-s + (0.503 − 0.863i)16-s + (0.700 + 0.713i)17-s + ⋯
L(s)  = 1  + (−0.966 + 0.257i)2-s + (−0.166 + 0.985i)3-s + (0.867 − 0.498i)4-s + (0.626 + 0.779i)5-s + (−0.0929 − 0.995i)6-s + (−0.616 + 0.787i)7-s + (−0.709 + 0.704i)8-s + (−0.944 − 0.329i)9-s + (−0.806 − 0.591i)10-s + (0.767 − 0.640i)11-s + (0.346 + 0.938i)12-s + (0.975 + 0.221i)13-s + (0.392 − 0.919i)14-s + (−0.873 + 0.487i)15-s + (0.503 − 0.863i)16-s + (0.700 + 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.772 + 0.634i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.772 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3116667808 + 0.8709808090i\)
\(L(\frac12)\) \(\approx\) \(0.3116667808 + 0.8709808090i\)
\(L(1)\) \(\approx\) \(0.6003113936 + 0.4906294694i\)
\(L(1)\) \(\approx\) \(0.6003113936 + 0.4906294694i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.966 + 0.257i)T \)
3 \( 1 + (-0.166 + 0.985i)T \)
5 \( 1 + (0.626 + 0.779i)T \)
7 \( 1 + (-0.616 + 0.787i)T \)
11 \( 1 + (0.767 - 0.640i)T \)
13 \( 1 + (0.975 + 0.221i)T \)
17 \( 1 + (0.700 + 0.713i)T \)
19 \( 1 + (0.999 + 0.0248i)T \)
29 \( 1 + (-0.556 + 0.831i)T \)
31 \( 1 + (-0.492 + 0.870i)T \)
37 \( 1 + (0.890 + 0.454i)T \)
41 \( 1 + (0.735 - 0.678i)T \)
43 \( 1 + (0.545 - 0.837i)T \)
47 \( 1 + (-0.334 + 0.942i)T \)
53 \( 1 + (-0.806 + 0.591i)T \)
59 \( 1 + (0.827 - 0.561i)T \)
61 \( 1 + (-0.999 - 0.0372i)T \)
67 \( 1 + (0.992 - 0.123i)T \)
71 \( 1 + (0.922 - 0.386i)T \)
73 \( 1 + (-0.556 - 0.831i)T \)
79 \( 1 + (-0.287 + 0.957i)T \)
83 \( 1 + (-0.820 - 0.571i)T \)
89 \( 1 + (-0.834 + 0.551i)T \)
97 \( 1 + (-0.514 - 0.857i)T \)
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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

−23.17811426085736208510016298301, −22.471315424500874438526935800291, −21.11394483925069137418596641156, −20.201469635337936222201917746927, −19.967398166143403176877421790844, −18.82150419982700530639718691493, −18.06602348220456250203712940599, −17.36098734510732613354626257864, −16.62317658601088788592765332475, −16.05522682862756793647159927187, −14.44066922917256811238000668588, −13.37435654386604545852381434790, −12.836992578334293014864795693618, −11.867377684803487445013401703579, −11.09904991749594488742797239667, −9.736884700444320743827687165951, −9.37670289974744045010164557441, −8.11717257416472955875508866869, −7.38692579017737034076741275109, −6.451496971114758648747914199067, −5.64963661042604305644980951087, −3.93197510061940484301374636312, −2.66329901380013427552708295416, −1.42353370546134903628549901534, −0.79649197692600037310814686155, 1.43587045504021682548734087645, 2.91267896405936096485651278772, 3.57449007710918768124653153714, 5.57886506980271570104833228889, 5.9517256583686712636311419029, 6.86082827737797673689189691368, 8.32038353380365376014907579327, 9.240028766217125066582400556, 9.63382250426948304321644993847, 10.753417108559515488073231912520, 11.2223331477931058010372279489, 12.32759698504054596405678896058, 13.998512725169578669112953893111, 14.610529302046877167404494767680, 15.561162289556503740791958330197, 16.233736007717496206362704240515, 16.94696772696919318842609150705, 17.9292515570021605125596965495, 18.66469652548883929802550974652, 19.40654392459539786408764536608, 20.45456138387383925881840874189, 21.37984998665807664547204176578, 21.98314619431703128452877621281, 22.79690467788686623189487919914, 23.89218926027467882298044755276

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