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Properties

Label 1-2664-2664.1363-r0-0-0
Degree $1$
Conductor $2664$
Sign $0.00469 - 0.999i$
Analytic cond. $12.3715$
Root an. cond. $12.3715$
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  + (0.866 − 0.5i)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s − i·17-s − i·19-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.866 + 0.5i)31-s − i·35-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s − i·17-s − i·19-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.866 + 0.5i)31-s − i·35-s + (0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.5 − 0.866i)47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00469 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00469 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.00469 - 0.999i$
Analytic conductor: \(12.3715\)
Root analytic conductor: \(12.3715\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (1363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2664,\ (0:\ ),\ 0.00469 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.766382431 - 1.774690022i\)
\(L(\frac12)\) \(\approx\) \(1.766382431 - 1.774690022i\)
\(L(1)\) \(\approx\) \(1.372776208 - 0.5322521132i\)
\(L(1)\) \(\approx\) \(1.372776208 - 0.5322521132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 - iT \)
19 \( 1 - iT \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−19.12036079400943297994765235167, −18.89840443692165663860271517793, −18.013032921878938258335767715579, −17.434469084061645949081382882343, −16.93145422455185461884695097167, −15.7793971369093025734200102035, −15.222219735173119536510809079871, −14.347313092829538888458399636124, −14.13669041516455420072330819573, −12.939032961079911495822919620971, −12.4831027862986491254630370843, −11.52605930375484759270836658879, −10.899070249666062524045004990761, −10.112842020238915007694498969924, −9.290583925213968427040470564312, −8.80410191462067532725137665438, −7.83176256242074759617698476368, −6.97969968244603073749773779563, −6.02344593542957554977914662875, −5.78574694874915221056812607296, −4.64944611480502380781021099129, −3.82098070699647417158316387455, −2.82569627987802659236938358951, −1.79405877476479143741865021213, −1.54064663794710485615933296089, 0.914178274836057461204262265511, 1.17000475798230121632656774398, 2.54590682213914660467419765622, 3.30601925785161356518528587980, 4.39351936620470391090841114634, 5.02304656785010401996367021096, 5.83322282399164938105185183726, 6.657323030963331929297736748220, 7.35415452305924157999303563802, 8.472733989029010354345460494069, 8.88363697988471527216513570843, 9.68536924495506663695115101613, 10.7028359639166754338951224287, 11.0165027269347288081844288130, 11.930062950959864106024411289144, 13.05755237470214472412182970208, 13.35589358882140892664366748127, 14.152488723605171009756884880219, 14.59559940450336301352088835776, 15.880411696455929182792362564648, 16.32932021507523304153970076821, 17.06239514496007555173808014830, 17.748426500740988504756933390039, 18.197233197653730697823916005632, 19.172310619432261037198595602573

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