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Properties

Label 1-1012-1012.251-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.133 - 0.991i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (0.0285 − 0.999i)3-s + (0.998 − 0.0570i)5-s + (0.897 − 0.441i)7-s + (−0.998 − 0.0570i)9-s + (0.696 − 0.717i)13-s + (−0.0285 − 0.999i)15-s + (−0.974 − 0.226i)17-s + (0.516 + 0.856i)19-s + (−0.415 − 0.909i)21-s + (0.993 − 0.113i)25-s + (−0.0855 + 0.996i)27-s + (0.516 − 0.856i)29-s + (−0.610 − 0.791i)31-s + (0.870 − 0.491i)35-s + (−0.774 − 0.633i)37-s + ⋯
L(s)  = 1  + (0.0285 − 0.999i)3-s + (0.998 − 0.0570i)5-s + (0.897 − 0.441i)7-s + (−0.998 − 0.0570i)9-s + (0.696 − 0.717i)13-s + (−0.0285 − 0.999i)15-s + (−0.974 − 0.226i)17-s + (0.516 + 0.856i)19-s + (−0.415 − 0.909i)21-s + (0.993 − 0.113i)25-s + (−0.0855 + 0.996i)27-s + (0.516 − 0.856i)29-s + (−0.610 − 0.791i)31-s + (0.870 − 0.491i)35-s + (−0.774 − 0.633i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.133 - 0.991i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.133 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.295076728 - 1.481413493i\)
\(L(\frac12)\) \(\approx\) \(1.295076728 - 1.481413493i\)
\(L(1)\) \(\approx\) \(1.225410806 - 0.6145653118i\)
\(L(1)\) \(\approx\) \(1.225410806 - 0.6145653118i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.0285 - 0.999i)T \)
5 \( 1 + (0.998 - 0.0570i)T \)
7 \( 1 + (0.897 - 0.441i)T \)
13 \( 1 + (0.696 - 0.717i)T \)
17 \( 1 + (-0.974 - 0.226i)T \)
19 \( 1 + (0.516 + 0.856i)T \)
29 \( 1 + (0.516 - 0.856i)T \)
31 \( 1 + (-0.610 - 0.791i)T \)
37 \( 1 + (-0.774 - 0.633i)T \)
41 \( 1 + (0.774 - 0.633i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.466 - 0.884i)T \)
59 \( 1 + (0.985 - 0.170i)T \)
61 \( 1 + (-0.941 + 0.336i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (0.870 + 0.491i)T \)
73 \( 1 + (-0.921 - 0.389i)T \)
79 \( 1 + (0.696 - 0.717i)T \)
83 \( 1 + (-0.362 + 0.931i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.362 + 0.931i)T \)
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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

−21.59727108787434392859289074990, −21.36716616611941487431912227565, −20.413494134337619543797331627900, −19.790707515996322678836499751067, −18.42352322431191220639211050419, −17.897936559371939227409162592027, −17.16284866518034660670177512049, −16.323179924331362222663001865660, −15.51137838168924937451600731432, −14.763211410076531634068693708423, −13.98771793123819316866465036980, −13.42009738646130604520568725984, −12.1205591815170872423289530689, −11.16639928757553025692307010686, −10.72799628712629262776158581827, −9.71068775726486355464796767248, −8.84510432933683510727831667121, −8.56010058334786382513977608169, −6.98199939052662941486469171406, −6.097674874936033403940637200352, −5.12711318842115743828773277041, −4.659984488998006391836899014461, −3.42674056279779372445702828940, −2.39585538266325406445853005015, −1.4736229761224296224744395319, 0.87021001503285724349889277253, 1.76055325666231233219249562694, 2.52428975147939818624081505838, 3.80462978392271849892062645343, 5.11439069263842986496822642070, 5.82572023844149912526324252428, 6.63406992703369085560058493489, 7.6039639537097821624019970872, 8.31241348912298800114337164215, 9.15094463485663205559642461563, 10.2766911649570413726848123250, 11.05761962301841335089719325389, 11.844934285808928975112361770678, 12.9275573713129631585088853169, 13.44389207060926600514130664534, 14.132577305961476080709774823223, 14.79814482576286233971211174447, 16.022982782789207460242478488121, 17.0335910806085335561975235899, 17.75306863561377101678836057070, 18.026106994523882845234518546966, 18.90446323429887306490608799703, 19.98311084749126676142467535696, 20.6205681261093686234494684188, 21.182517283238200437610942433240

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