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Properties

Label 1-23e2-529.381-r0-0-0
Degree $1$
Conductor $529$
Sign $0.999 + 0.0118i$
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  + (−0.977 − 0.209i)2-s + (−0.935 + 0.352i)3-s + (0.912 + 0.409i)4-s + (0.984 + 0.172i)5-s + (0.988 − 0.148i)6-s + (−0.907 − 0.421i)7-s + (−0.806 − 0.591i)8-s + (0.751 − 0.659i)9-s + (−0.926 − 0.375i)10-s + (−0.885 + 0.465i)11-s + (−0.998 − 0.0620i)12-s + (−0.0434 + 0.999i)13-s + (0.798 + 0.601i)14-s + (−0.982 + 0.185i)15-s + (0.664 + 0.747i)16-s + (0.323 − 0.946i)17-s + ⋯
L(s)  = 1  + (−0.977 − 0.209i)2-s + (−0.935 + 0.352i)3-s + (0.912 + 0.409i)4-s + (0.984 + 0.172i)5-s + (0.988 − 0.148i)6-s + (−0.907 − 0.421i)7-s + (−0.806 − 0.591i)8-s + (0.751 − 0.659i)9-s + (−0.926 − 0.375i)10-s + (−0.885 + 0.465i)11-s + (−0.998 − 0.0620i)12-s + (−0.0434 + 0.999i)13-s + (0.798 + 0.601i)14-s + (−0.982 + 0.185i)15-s + (0.664 + 0.747i)16-s + (0.323 − 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.999 + 0.0118i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.999 + 0.0118i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6219372499 + 0.003693566098i\)
\(L(\frac12)\) \(\approx\) \(0.6219372499 + 0.003693566098i\)
\(L(1)\) \(\approx\) \(0.5748241343 + 0.007620089448i\)
\(L(1)\) \(\approx\) \(0.5748241343 + 0.007620089448i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.977 - 0.209i)T \)
3 \( 1 + (-0.935 + 0.352i)T \)
5 \( 1 + (0.984 + 0.172i)T \)
7 \( 1 + (-0.907 - 0.421i)T \)
11 \( 1 + (-0.885 + 0.465i)T \)
13 \( 1 + (-0.0434 + 0.999i)T \)
17 \( 1 + (0.323 - 0.946i)T \)
19 \( 1 + (0.346 - 0.938i)T \)
29 \( 1 + (0.586 + 0.809i)T \)
31 \( 1 + (-0.117 - 0.993i)T \)
37 \( 1 + (-0.426 + 0.904i)T \)
41 \( 1 + (0.369 - 0.929i)T \)
43 \( 1 + (-0.0186 - 0.999i)T \)
47 \( 1 + (0.854 + 0.519i)T \)
53 \( 1 + (-0.926 + 0.375i)T \)
59 \( 1 + (-0.596 - 0.802i)T \)
61 \( 1 + (0.251 + 0.967i)T \)
67 \( 1 + (0.980 - 0.197i)T \)
71 \( 1 + (0.813 + 0.581i)T \)
73 \( 1 + (0.586 - 0.809i)T \)
79 \( 1 + (-0.986 + 0.160i)T \)
83 \( 1 + (0.0310 - 0.999i)T \)
89 \( 1 + (0.948 + 0.317i)T \)
97 \( 1 + (-0.972 + 0.233i)T \)
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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

−23.51122287758133952910035084798, −22.809197406096030121290485594884, −21.646621450524440831025076105896, −21.118492273737366125822802416938, −19.869686289558818714398976984866, −18.96105001549219959455057891301, −18.27592636945252656217888244088, −17.65556514785026732407973127951, −16.822729444233816690533795651420, −16.1346436813619127717683672041, −15.41204226960055050275528210002, −14.04273750871654838985884970403, −12.7911522704108623641256621459, −12.45855213580399135949148500140, −11.06568085925479284796476503316, −10.21829140380121162482313239682, −9.85121924417858954830349813252, −8.51675127829773517570403055279, −7.67111184326021427828561444759, −6.42332506371815471419175271940, −5.87593526625734318023024404561, −5.265934307965065127085015339968, −3.11057608759956606997461830114, −2.0108851903341716935543722912, −0.81351807288670364807314449714, 0.75416144438266889804355340755, 2.16343393956664805743395855742, 3.24043670040564262838965284421, 4.74002327170250477790304818984, 5.84412030727418437113189284833, 6.816686898223618775440713217527, 7.27746941267119536294332515894, 9.09235783540597946754220063150, 9.63700116024939651536011979145, 10.32592125735650081143714321269, 11.06329921039911053980102255712, 12.10503476292747933491639322728, 12.948970083429159980989593187296, 13.96267560699829156820382902969, 15.49295090996141022236769955373, 16.06093568191537029157758513741, 16.93255673896961060225669187875, 17.462974370325063215430884121919, 18.41992437048088478232291509835, 18.88643298783567963202335751277, 20.29408280365896015633490239139, 20.86357035615700922769036326561, 21.8118686603821723440294743782, 22.38686953490774616813651378189, 23.53087859286411386052567459317

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