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Properties

Label 1-133-133.103-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: $\chi_{133} (103, \cdot )$
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

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

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