Properties

Label 1-23e2-529.376-r0-0-0
Degree $1$
Conductor $529$
Sign $0.984 + 0.176i$
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  + (0.767 + 0.640i)2-s + (0.901 − 0.432i)3-s + (0.179 + 0.983i)4-s + (−0.726 − 0.687i)5-s + (0.969 + 0.245i)6-s + (0.948 + 0.317i)7-s + (−0.492 + 0.870i)8-s + (0.626 − 0.779i)9-s + (−0.117 − 0.993i)10-s + (−0.691 − 0.722i)11-s + (0.586 + 0.809i)12-s + (0.827 − 0.561i)13-s + (0.524 + 0.851i)14-s + (−0.952 − 0.305i)15-s + (−0.935 + 0.352i)16-s + (0.999 − 0.0248i)17-s + ⋯
L(s)  = 1  + (0.767 + 0.640i)2-s + (0.901 − 0.432i)3-s + (0.179 + 0.983i)4-s + (−0.726 − 0.687i)5-s + (0.969 + 0.245i)6-s + (0.948 + 0.317i)7-s + (−0.492 + 0.870i)8-s + (0.626 − 0.779i)9-s + (−0.117 − 0.993i)10-s + (−0.691 − 0.722i)11-s + (0.586 + 0.809i)12-s + (0.827 − 0.561i)13-s + (0.524 + 0.851i)14-s + (−0.952 − 0.305i)15-s + (−0.935 + 0.352i)16-s + (0.999 − 0.0248i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.984 + 0.176i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.984 + 0.176i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.775562736 + 0.2463958463i\)
\(L(\frac12)\) \(\approx\) \(2.775562736 + 0.2463958463i\)
\(L(1)\) \(\approx\) \(2.009313659 + 0.2551624075i\)
\(L(1)\) \(\approx\) \(2.009313659 + 0.2551624075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.767 + 0.640i)T \)
3 \( 1 + (0.901 - 0.432i)T \)
5 \( 1 + (-0.726 - 0.687i)T \)
7 \( 1 + (0.948 + 0.317i)T \)
11 \( 1 + (-0.691 - 0.722i)T \)
13 \( 1 + (0.827 - 0.561i)T \)
17 \( 1 + (0.999 - 0.0248i)T \)
19 \( 1 + (-0.556 - 0.831i)T \)
29 \( 1 + (0.867 + 0.498i)T \)
31 \( 1 + (0.751 + 0.659i)T \)
37 \( 1 + (-0.977 + 0.209i)T \)
41 \( 1 + (-0.404 + 0.914i)T \)
43 \( 1 + (0.0310 - 0.999i)T \)
47 \( 1 + (-0.990 - 0.136i)T \)
53 \( 1 + (-0.117 + 0.993i)T \)
59 \( 1 + (-0.0186 + 0.999i)T \)
61 \( 1 + (0.995 - 0.0991i)T \)
67 \( 1 + (-0.191 - 0.981i)T \)
71 \( 1 + (-0.999 - 0.0124i)T \)
73 \( 1 + (0.867 - 0.498i)T \)
79 \( 1 + (0.251 + 0.967i)T \)
83 \( 1 + (0.890 + 0.454i)T \)
89 \( 1 + (-0.873 - 0.487i)T \)
97 \( 1 + (0.130 + 0.991i)T \)
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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

−23.3125991452995060971767388287, −22.71025573406954473217772374600, −21.47986565645439896996956264810, −20.85700891567829218967532018973, −20.49434057715068284677813284266, −19.226130283199953592306355141502, −18.910021103202654352422347710266, −17.86568802459240922862115829822, −16.22542150417350618584843649260, −15.48438711365293443217424742959, −14.66109059194349972793203621170, −14.24933515715805192102551141580, −13.33276015071337391512779603295, −12.209218464746394518804633400443, −11.33287380355271621467238215631, −10.44583794356343495933852802757, −9.93059238949878089365791025918, −8.41872067759731131774387471565, −7.721404581063367965675733314401, −6.60489904914114381496205136178, −5.14441637069097221955325531631, −4.22783532466432702158011634092, −3.61621527528357156012819392657, −2.517668454218434198416119845594, −1.55910325437914872139352806237, 1.24149262715727752375322020465, 2.77145973220113671619006716260, 3.55589447681003658229439217071, 4.68443953468048398476122708937, 5.48756733366183692405040054049, 6.75670403835536657932841118098, 7.89529042439158332777578504380, 8.29189135254305775828208224698, 8.8713883395066994229856232130, 10.700402648968873240558749949352, 11.819053883346753226343688174058, 12.49436498593493121676329158073, 13.40092815284919100371247141851, 14.020239724387919826325797256498, 15.10071095735108080395469506844, 15.5564959713228558022003581334, 16.42318780017517560998028486162, 17.59197344994461424905026167202, 18.402422973903238703305698478009, 19.40738045174385653935786574242, 20.4272441190662477600722703341, 21.04343653819707728968689321280, 21.5548436436048499276745806939, 23.11739536513870705492010666040, 23.71173376246444233319316204535

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