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Properties

Label 1-133-133.31-r0-0-0
Degree $1$
Conductor $133$
Sign $0.0288 + 0.999i$
Analytic cond. $0.617649$
Root an. cond. $0.617649$
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  + (0.5 + 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s − 17-s + (0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s − 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s − 17-s + (0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0288 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0288 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.0288 + 0.999i$
Analytic conductor: \(0.617649\)
Root analytic conductor: \(0.617649\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (0:\ ),\ 0.0288 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248955596 + 1.285528317i\)
\(L(\frac12)\) \(\approx\) \(1.248955596 + 1.285528317i\)
\(L(1)\) \(\approx\) \(1.394229802 + 0.9090168389i\)
\(L(1)\) \(\approx\) \(1.394229802 + 0.9090168389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−28.737232966103817494567013593511, −27.48250526445933797222431797445, −26.47685534040182901975629692322, −25.232192737578050904884994705980, −24.35462150566334524483298330293, −23.44607916123340800868136861439, −21.953045116585383109926443101581, −21.15485499522572046055202229788, −20.38224514215742244687433589022, −19.644165517526482874871984296848, −18.54961102481697576550477733644, −17.36518151198128481996678225702, −15.7289812994279198746038906578, −14.72867975959994827627815044639, −13.61577298756888118453910728447, −12.97318117343876539198818403368, −11.95237852515567599608461087602, −10.3088244285091169835014326112, −9.41259918739399448883916355255, −8.59397740301799342332535447605, −6.85402479994754644356583495511, −5.02500261456206780090043078874, −4.239122564440049660442283047422, −2.58165142724343391538584359241, −1.65315025524722520524032260807, 2.58585966436758615153810747877, 3.42498885478958889646939429173, 5.036790200703310938544529197249, 6.414258281664336842708803690660, 7.452061152989905368495646240482, 8.4771237280383581282065100756, 9.64524395936750730610936301143, 11.015750327195939750838385620157, 12.929496774565146191805208524115, 13.49781163913651657605000242675, 14.62897291109803964539466320889, 15.18337909581059148828801784190, 16.34958745959061404433324591405, 17.78200198284479485199911828715, 18.52969359554203048524545291335, 19.81119102031229967022633888118, 21.19074454240958238388788222858, 21.84200544076799875603613442371, 22.873052376444366677833030111474, 24.179290467904917355124855250517, 24.95677651286324295723529460417, 25.79744761468493708742384631106, 26.65525231762361963524169828531, 27.20534139006429004357777474164, 29.295424279502563374783224307936

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