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Properties

Label 1-2667-2667.920-r0-0-0
Degree $1$
Conductor $2667$
Sign $-0.326 + 0.945i$
Analytic cond. $12.3854$
Root an. cond. $12.3854$
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  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (0.826 + 0.563i)5-s + (−0.623 + 0.781i)8-s + (0.222 + 0.974i)10-s + (0.998 − 0.0498i)11-s + (0.411 − 0.911i)13-s + (−0.988 + 0.149i)16-s + (−0.969 − 0.246i)17-s − 19-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (0.998 + 0.0498i)23-s + (0.365 + 0.930i)25-s + (0.921 − 0.388i)26-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (0.826 + 0.563i)5-s + (−0.623 + 0.781i)8-s + (0.222 + 0.974i)10-s + (0.998 − 0.0498i)11-s + (0.411 − 0.911i)13-s + (−0.988 + 0.149i)16-s + (−0.969 − 0.246i)17-s − 19-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (0.998 + 0.0498i)23-s + (0.365 + 0.930i)25-s + (0.921 − 0.388i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.326 + 0.945i$
Analytic conductor: \(12.3854\)
Root analytic conductor: \(12.3854\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (920, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (0:\ ),\ -0.326 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.836437103 + 2.576314208i\)
\(L(\frac12)\) \(\approx\) \(1.836437103 + 2.576314208i\)
\(L(1)\) \(\approx\) \(1.560767274 + 1.015520336i\)
\(L(1)\) \(\approx\) \(1.560767274 + 1.015520336i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.733 + 0.680i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (0.998 - 0.0498i)T \)
13 \( 1 + (0.411 - 0.911i)T \)
17 \( 1 + (-0.969 - 0.246i)T \)
19 \( 1 - T \)
23 \( 1 + (0.998 + 0.0498i)T \)
29 \( 1 + (0.0249 - 0.999i)T \)
31 \( 1 + (0.969 - 0.246i)T \)
37 \( 1 + (0.766 + 0.642i)T \)
41 \( 1 + (0.270 + 0.962i)T \)
43 \( 1 + (0.878 - 0.478i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.124 + 0.992i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (-0.661 + 0.749i)T \)
71 \( 1 + (-0.542 + 0.840i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (0.542 - 0.840i)T \)
83 \( 1 + (-0.797 + 0.603i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (0.853 - 0.521i)T \)
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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

−19.41793665314197889755185577205, −18.526208059507334801126771023109, −17.69371994749168188168866445127, −17.01895933504650914491542017607, −16.27139770592729310905236253491, −15.419565736421128524990089679545, −14.50329209719516612097475446251, −14.13048426934311832238733714984, −13.24623037808068426912674721599, −12.81192242381967322189808644253, −12.03672046552234638919223034575, −11.222081773380672579830771376478, −10.65059193607395181345447822807, −9.74149277773027896889700772459, −8.94089278229836717286936268766, −8.72551527890486996469333083771, −6.96097138260648569482817395100, −6.449470589788266699492956087521, −5.797449483848211702413608810970, −4.72356907216807252455097857837, −4.34136064331907222303706089166, −3.40493853696997226883089987753, −2.2455256735877836905360105923, −1.74934892092339634003702237894, −0.82213838602020946933528487380, 1.174830008615912553971663651892, 2.50220605981800940764921221322, 2.90528872188871911671295813845, 4.08437709794653544706142763159, 4.62749696943649191327009067952, 5.82186263775523110436417160971, 6.17599305940055384738664983475, 6.85303270288035817630041868740, 7.66757628982832041816125480822, 8.645335367802787868155499968059, 9.19494517352597805746749933843, 10.2109089659871597727160107691, 11.07272288971233121190402754414, 11.65433603635014500574367015956, 12.6986454913904718160078713289, 13.34523406800244735316863662492, 13.734753854840726697209860112214, 14.791799281339904356435575808432, 14.99625623530897367750878289028, 15.83905864345504018985433465574, 16.79325921005790666436720108464, 17.42639219676330085921244163961, 17.721428617691280823161142112686, 18.72633266377164193624969731462, 19.52173613100053693449331624639

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