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Properties

Label 1-668-668.155-r0-0-0
Degree $1$
Conductor $668$
Sign $0.747 + 0.664i$
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  + (0.982 + 0.188i)3-s + (−0.954 − 0.298i)5-s + (0.387 + 0.922i)7-s + (0.929 + 0.369i)9-s + (0.584 − 0.811i)11-s + (−0.974 + 0.225i)13-s + (−0.881 − 0.472i)15-s + (0.942 + 0.334i)17-s + (−0.280 + 0.959i)19-s + (0.206 + 0.978i)21-s + (0.132 − 0.991i)23-s + (0.822 + 0.569i)25-s + (0.843 + 0.537i)27-s + (0.132 + 0.991i)29-s + (0.521 − 0.853i)31-s + ⋯
L(s)  = 1  + (0.982 + 0.188i)3-s + (−0.954 − 0.298i)5-s + (0.387 + 0.922i)7-s + (0.929 + 0.369i)9-s + (0.584 − 0.811i)11-s + (−0.974 + 0.225i)13-s + (−0.881 − 0.472i)15-s + (0.942 + 0.334i)17-s + (−0.280 + 0.959i)19-s + (0.206 + 0.978i)21-s + (0.132 − 0.991i)23-s + (0.822 + 0.569i)25-s + (0.843 + 0.537i)27-s + (0.132 + 0.991i)29-s + (0.521 − 0.853i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(3.10217\)
Root analytic conductor: \(3.10217\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{668} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 668,\ (0:\ ),\ 0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.696899625 + 0.6453030359i\)
\(L(\frac12)\) \(\approx\) \(1.696899625 + 0.6453030359i\)
\(L(1)\) \(\approx\) \(1.342388901 + 0.2223421118i\)
\(L(1)\) \(\approx\) \(1.342388901 + 0.2223421118i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (0.982 + 0.188i)T \)
5 \( 1 + (-0.954 - 0.298i)T \)
7 \( 1 + (0.387 + 0.922i)T \)
11 \( 1 + (0.584 - 0.811i)T \)
13 \( 1 + (-0.974 + 0.225i)T \)
17 \( 1 + (0.942 + 0.334i)T \)
19 \( 1 + (-0.280 + 0.959i)T \)
23 \( 1 + (0.132 - 0.991i)T \)
29 \( 1 + (0.132 + 0.991i)T \)
31 \( 1 + (0.521 - 0.853i)T \)
37 \( 1 + (-0.929 + 0.369i)T \)
41 \( 1 + (0.455 + 0.890i)T \)
43 \( 1 + (0.351 + 0.936i)T \)
47 \( 1 + (0.644 - 0.764i)T \)
53 \( 1 + (0.0189 - 0.999i)T \)
59 \( 1 + (-0.942 + 0.334i)T \)
61 \( 1 + (0.421 + 0.906i)T \)
67 \( 1 + (0.954 - 0.298i)T \)
71 \( 1 + (0.489 + 0.872i)T \)
73 \( 1 + (0.243 - 0.969i)T \)
79 \( 1 + (0.672 + 0.739i)T \)
83 \( 1 + (0.898 + 0.438i)T \)
89 \( 1 + (-0.881 + 0.472i)T \)
97 \( 1 + (-0.521 - 0.853i)T \)
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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

−22.888121071471152945563998214391, −21.81270570823461128991758596291, −20.78675543400303501996859807682, −20.12088044074972445709781674921, −19.46565448005806128930068434129, −18.9855934522184948789643217576, −17.66250229972080634784314616616, −17.12720631808156368967967618651, −15.730251569657090652978321516430, −15.23240408897767707477526298357, −14.29725031360916163311974487661, −13.84848216687298268103148003200, −12.5432225440919211592974421717, −11.98939727083195324566800887703, −10.820106677089665016768188239019, −9.92768192207288859108770672455, −9.07448783219231738775141907704, −7.89181064544819833771769299779, −7.38812312988069888340201087106, −6.82106618961491008353924604498, −4.94116393771523628097312855197, −4.13224862903484096462709330665, −3.330138955267493826035720825583, −2.236704848970533148962148422562, −0.914959892748203708071340293527, 1.3091455601794638175370127143, 2.553011358500732177238678435956, 3.47570297649997548478410195160, 4.36794259206940831512545262667, 5.33405302748544024535930565547, 6.64929453033700004517242155687, 7.8553274417263965021100633473, 8.32825366086098863225486576774, 9.08127130862328960433890509149, 10.050856790830850713257341821503, 11.18705356051049454289226952585, 12.20727005608481833495532960681, 12.60360973224065789943614072015, 14.03186756729459694177662118949, 14.711772453694962770397426205337, 15.170654925261036639275615411636, 16.32013193340547097862294176066, 16.75932184485859231348107657182, 18.30543312588877628510703919407, 19.09960896158677700275926993565, 19.39367359642626694058165003944, 20.458078243379901686480273139062, 21.16297580813439775798104697779, 21.89806969479788005182641776347, 22.792214383748420237352929408722

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