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Properties

Label 1-2672-2672.1435-r1-0-0
Degree $1$
Conductor $2672$
Sign $0.0597 - 0.998i$
Analytic cond. $287.146$
Root an. cond. $287.146$
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  + (−0.922 − 0.387i)3-s + (−0.948 − 0.316i)5-s + (−0.965 − 0.261i)7-s + (0.700 + 0.713i)9-s + (0.404 + 0.914i)11-s + (0.150 − 0.988i)13-s + (0.752 + 0.658i)15-s + (0.974 − 0.225i)17-s + (−0.188 − 0.982i)19-s + (0.788 + 0.614i)21-s + (0.421 − 0.906i)23-s + (0.800 + 0.599i)25-s + (−0.369 − 0.929i)27-s + (0.906 − 0.421i)29-s + (−0.776 − 0.629i)31-s + ⋯
L(s)  = 1  + (−0.922 − 0.387i)3-s + (−0.948 − 0.316i)5-s + (−0.965 − 0.261i)7-s + (0.700 + 0.713i)9-s + (0.404 + 0.914i)11-s + (0.150 − 0.988i)13-s + (0.752 + 0.658i)15-s + (0.974 − 0.225i)17-s + (−0.188 − 0.982i)19-s + (0.788 + 0.614i)21-s + (0.421 − 0.906i)23-s + (0.800 + 0.599i)25-s + (−0.369 − 0.929i)27-s + (0.906 − 0.421i)29-s + (−0.776 − 0.629i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0597 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0597 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2672\)    =    \(2^{4} \cdot 167\)
Sign: $0.0597 - 0.998i$
Analytic conductor: \(287.146\)
Root analytic conductor: \(287.146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2672} (1435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2672,\ (1:\ ),\ 0.0597 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8830329877 - 0.8317331356i\)
\(L(\frac12)\) \(\approx\) \(0.8830329877 - 0.8317331356i\)
\(L(1)\) \(\approx\) \(0.6616407683 - 0.2163904712i\)
\(L(1)\) \(\approx\) \(0.6616407683 - 0.2163904712i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 \)
good3 \( 1 + (-0.922 - 0.387i)T \)
5 \( 1 + (-0.948 - 0.316i)T \)
7 \( 1 + (-0.965 - 0.261i)T \)
11 \( 1 + (0.404 + 0.914i)T \)
13 \( 1 + (0.150 - 0.988i)T \)
17 \( 1 + (0.974 - 0.225i)T \)
19 \( 1 + (-0.188 - 0.982i)T \)
23 \( 1 + (0.421 - 0.906i)T \)
29 \( 1 + (0.906 - 0.421i)T \)
31 \( 1 + (-0.776 - 0.629i)T \)
37 \( 1 + (-0.713 - 0.700i)T \)
41 \( 1 + (-0.206 - 0.978i)T \)
43 \( 1 + (0.959 - 0.280i)T \)
47 \( 1 + (-0.0567 + 0.998i)T \)
53 \( 1 + (0.872 + 0.489i)T \)
59 \( 1 + (0.225 - 0.974i)T \)
61 \( 1 + (-0.686 - 0.726i)T \)
67 \( 1 + (0.948 - 0.316i)T \)
71 \( 1 + (-0.942 - 0.334i)T \)
73 \( 1 + (-0.351 + 0.936i)T \)
79 \( 1 + (0.881 + 0.472i)T \)
83 \( 1 + (0.298 + 0.954i)T \)
89 \( 1 + (0.752 - 0.658i)T \)
97 \( 1 + (0.776 - 0.629i)T \)
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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

−19.14512878757481309265415496794, −18.76779096339442199646864998109, −17.98075455288272341432511169069, −16.85011409571224822716695667991, −16.34317073068631557598833734324, −16.125529243708096561371901536119, −15.14794312972453131100108450496, −14.52746786871728216179696267854, −13.583627278754006331094026444292, −12.62060867112330772308676234973, −11.88406942494396485201702727946, −11.670866790919862567408292768044, −10.6364549348548492508301155829, −10.14882440194817153325999927277, −9.18164564307803266621818302923, −8.54304597274845784287347690325, −7.42228292846300767500083576258, −6.72189506954318978833064436005, −6.095921388772059441414241303908, −5.367058821916828640267034570201, −4.31488420837913691914310718398, −3.53066272842584797054073189958, −3.19210426988445271270617775217, −1.51045510002196742564387564183, −0.61201696812841085866124240554, 0.55301052592514605091209422823, 0.7689722791917119191168712187, 2.235966110820314224707719188227, 3.27761806217980951155702792275, 4.14043008108805694171463442996, 4.86535356493845185521155949293, 5.678373217420403111806422270334, 6.58182006047220295892545606679, 7.241629987567149715151121031007, 7.73075642831512331223339947136, 8.80324760898994860542216836034, 9.67028749374357922796666734887, 10.49383810648044324140203025917, 11.03214844614993600130138400729, 12.02851765578765435763272385041, 12.54473992360649390493403460007, 12.84468346952248962754170549534, 13.85057647974231200850601758448, 14.8961076899030438712245514224, 15.74098376598932741989720853862, 16.00960184316752471876189811288, 17.03948359019380472334905543735, 17.3035504235900735548318250278, 18.31407561286974843575722541720, 18.999842341776152479246961480028

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