Properties

Label 1-23e2-529.361-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: $\chi_{529} (361, \cdot )$
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

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

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