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Properties

Label 1-668-668.111-r0-0-0
Degree $1$
Conductor $668$
Sign $0.551 - 0.834i$
Analytic cond. $3.10217$
Root an. cond. $3.10217$
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  + (−0.132 + 0.991i)3-s + (0.914 − 0.404i)5-s + (−0.421 − 0.906i)7-s + (−0.965 − 0.261i)9-s + (−0.614 − 0.788i)11-s + (0.455 + 0.890i)13-s + (0.280 + 0.959i)15-s + (−0.997 + 0.0756i)17-s + (−0.553 − 0.832i)19-s + (0.954 − 0.298i)21-s + (0.929 − 0.369i)23-s + (0.672 − 0.739i)25-s + (0.387 − 0.922i)27-s + (0.929 + 0.369i)29-s + (−0.974 − 0.225i)31-s + ⋯
L(s)  = 1  + (−0.132 + 0.991i)3-s + (0.914 − 0.404i)5-s + (−0.421 − 0.906i)7-s + (−0.965 − 0.261i)9-s + (−0.614 − 0.788i)11-s + (0.455 + 0.890i)13-s + (0.280 + 0.959i)15-s + (−0.997 + 0.0756i)17-s + (−0.553 − 0.832i)19-s + (0.954 − 0.298i)21-s + (0.929 − 0.369i)23-s + (0.672 − 0.739i)25-s + (0.387 − 0.922i)27-s + (0.929 + 0.369i)29-s + (−0.974 − 0.225i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.551 - 0.834i$
Analytic conductor: \(3.10217\)
Root analytic conductor: \(3.10217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{668} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 668,\ (0:\ ),\ 0.551 - 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9975488661 - 0.5365446537i\)
\(L(\frac12)\) \(\approx\) \(0.9975488661 - 0.5365446537i\)
\(L(1)\) \(\approx\) \(0.9975013045 - 0.03708095035i\)
\(L(1)\) \(\approx\) \(0.9975013045 - 0.03708095035i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (-0.132 + 0.991i)T \)
5 \( 1 + (0.914 - 0.404i)T \)
7 \( 1 + (-0.421 - 0.906i)T \)
11 \( 1 + (-0.614 - 0.788i)T \)
13 \( 1 + (0.455 + 0.890i)T \)
17 \( 1 + (-0.997 + 0.0756i)T \)
19 \( 1 + (-0.553 - 0.832i)T \)
23 \( 1 + (0.929 - 0.369i)T \)
29 \( 1 + (0.929 + 0.369i)T \)
31 \( 1 + (-0.974 - 0.225i)T \)
37 \( 1 + (0.965 - 0.261i)T \)
41 \( 1 + (-0.898 - 0.438i)T \)
43 \( 1 + (-0.0944 - 0.995i)T \)
47 \( 1 + (0.0189 - 0.999i)T \)
53 \( 1 + (0.169 - 0.985i)T \)
59 \( 1 + (0.997 + 0.0756i)T \)
61 \( 1 + (-0.700 - 0.713i)T \)
67 \( 1 + (-0.914 - 0.404i)T \)
71 \( 1 + (-0.993 - 0.113i)T \)
73 \( 1 + (0.800 + 0.599i)T \)
79 \( 1 + (0.351 + 0.936i)T \)
83 \( 1 + (-0.584 - 0.811i)T \)
89 \( 1 + (0.280 - 0.959i)T \)
97 \( 1 + (0.974 - 0.225i)T \)
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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

−22.88423526839478808253107290709, −22.26866578876803667567173433618, −21.32885249594190764190456360906, −20.43785715413140014010345241136, −19.489212745822459054444366373061, −18.59375835471087190111530953102, −18.02168884393050690652671868193, −17.549243136022702939611006122491, −16.43609530940436250032046923238, −15.25959856263014603191259961444, −14.681036680971840934918342892180, −13.39183970002082234944751098534, −13.03465373641142799747912813474, −12.272540828923984493582494105374, −11.14157867764886074581870248081, −10.31084968692725519532122759312, −9.276941120653718173279509562205, −8.39156188529890514958627951312, −7.38421232827386318204885432554, −6.38274279141507252945759431509, −5.86655974261047145378914348509, −4.90441103727238947974449335084, −3.03373504921098457641075970999, −2.42110136503606813769909893862, −1.43683263408497705925990227901, 0.551959573132717282644469931941, 2.19272106410606950337631254745, 3.33345383351480626256753179030, 4.36960031838064449837190813993, 5.1112329206110596122021588616, 6.19179604128159579350906029853, 6.92210900708435539487385431359, 8.653270274937553402693163114418, 8.973259574908729479016143542942, 10.075298298626294786183187879309, 10.732969958749587202213308248294, 11.39448141742913630304505088852, 12.92142021939783207592210151555, 13.51903811824768443376859358884, 14.239290449974958778381878376903, 15.348223519419255495548202655285, 16.27255955871905932015147935238, 16.74159151474715199119440219946, 17.46822995736823620819059749450, 18.46037364269481228700022643617, 19.6436926001731770559904999405, 20.3427192108836346620916074391, 21.19225947913597462425135814830, 21.664549958501543125686529128246, 22.44976887121945905907803635057

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